Methods and systems for estimating time remaining and right sizing usable capacities of resources of a distributed computing system

ABSTRACT

Computational methods and systems that estimate time remaining and right size for usable capacities of resources used to run virtual objects of a distributed computing system are described. For each stream of metric data that represents usage of a resource of a distributed computing system, a model for forecasting metric data is determined and used to compute forecasted metric data in a forecast interval. A resource utilization metric is computed from the forecasted metric data and may be used to estimate a time remaining before the usable capacity of the resource is expected to be insufficient and the resource usable capacity is adjusted. The resource utilization metric may be used to determine the capacity remaining is insufficient. A right-size usable capacity for the resource is computed based on the resource utilization metric and the usable capacity of the resource is adjusted to at least the right-size usable capacity.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of patent application Ser. No. 16/013,503 filed Jun. 20, 2018 which is a continuation-in-part of patent application Ser. No. 15/951,523 filed Apr. 12, 2018.

TECHNICAL FIELD

This disclosure is directed to methods and systems to estimate time and/or capacity remaining before resources of distributed computing system become overburdened.

BACKGROUND

Electronic computing has evolved from primitive, vacuum-tube-based computer systems, initially developed during the 1940s, to modern electronic computing systems in which large numbers of multi-processor computer systems, such as server computers, work stations, and other individual computing systems are networked together with large-capacity data-storage devices and other electronic devices to produce geographically distributed computing systems with numerous components that provide enormous computational bandwidths and data-storage capacities. These large, distributed computing systems are made possible by advances in computer networking, distributed operating systems and applications, data-storage appliances, computer hardware, and software technologies.

Because distributed computing systems have such a large number of components, identification of performance problems is challenging and time consuming. Automated management systems have been developed to collected hundreds of thousands of streams of metric data generated by computational resources, such as processors, memory, storage devices, and network devices. Each data point of a stream of metric data represents an amount of the resource in use at a point in time. Management systems have historically used reactive monitoring to generate an alert when metric data of a corresponding resource violates a usage limit. Although reactive monitoring techniques are useful for identifying current performance problems, reactive monitoring techniques have scalability limitations and force system administrators to react immediately to performance problems that have already started to impact the performance of computational resources or are imminent.

In recent years, management systems have been developed to proactively identify performance problems by making short-term projections of resource usage and plan for changes in resource capacity based on historical metric data. For example, typical management systems use historical metric data of a resource (e.g., over past 3 to 6 months) to determine the time remaining before the resource is expected to be overburdened in relatively short time windows (e.g., an hour or a day). Capacity planning includes planning acquisition and provisioning of computational resources to satisfy future demand and reclaim underutilized resources. Typical management systems include capacity planning techniques that enable a system administrator to define peak resource usage by selecting a time window of historical metric data to assess past resource usage and determine a threshold above which future demand would be considered peak usage or stress on a resource. For example, a mean or maximum peak resource usage discovered in historical metric data may be used to set a threshold and drive capacity planning recommendations for the resource. However, determining the time remaining and capacity planning in this manner leads to problems: 1) Determination of time remaining and capacity planning is centered around finding historical peaks in resource usage and fails to account for trends and seasonality in resource usage over time. An online retailer, for example, may experience seasonal usage of resources that gradually increases over time. Such long-term changes in resource usage is not accounted for in typical management systems. 2) The computation involves finding arbitrarily user-defined peaks in batch operations, which can be computationally intensive, time consuming, and limited in usage. Distributed computing system administrators continue to seek methods and systems to estimate time remaining before resources are overburdened and plan for future demand in the capacity of resources.

SUMMARY

Computational methods and systems that estimate time remaining and right size for usable capacities of resources used to run virtual objects of a distributed computing system are described. Streams of metric data representing usage of various resources of the distributed computing system are sent to a management system of the distributed computing system. For each stream of metric data, the management system computes two or more models for forecasting metric data. The management system selects the model that corresponds to the behavior of the sequence of metric data and uses the model to compute forecasted metric data in a forecast interval. A resource utilization metric is computed from the forecasted metric data and may be used to estimate a time remaining before the usable capacity of the resource is expected to be insufficient and the usable capacity of the resource is adjusted. The resource utilization metric may be used to determine the capacity remaining of the resource. A right-size usable capacity for the resource is computed from the capacity remaining and the usable capacity of the resource is adjusted to at least the right-size usable capacity.

DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an architectural diagram for various types of computers.

FIG. 2 shows an Internet-connected distributed computer system.

FIG. 3 shows cloud computing.

FIG. 4 shows generalized hardware and software components of a general-purpose computer system.

FIGS. 5A-5B show two types of virtual machine (“VM”) and VM execution environments.

FIG. 6 shows an example of an open virtualization format package.

FIG. 7 shows virtual data centers provided as an abstraction of underlying physical-data-center hardware components.

FIG. 8 shows virtual-machine components of a virtual-data-center management server and physical servers of a physical data center.

FIG. 9 shows a cloud-director level of abstraction.

FIG. 10 shows virtual-cloud-connector nodes.

FIG. 11 shows an example server computer used to host three containers.

FIG. 12 shows an approach to implementing the containers on a VM.

FIG. 13 shows an example of a virtualization layer located above a physical data center.

FIG. 14A shows an example of a virtual infrastructure formed from resources of a server computer used to run two virtual objects.

FIG. 14B shows an example of a virtual infrastructure formed from resources of a cluster of server computers used to run the four virtual objects.

FIG. 15 shows a virtual infrastructure management tool receiving streams of metric data.

FIG. 16A-16D show plots of four different example streams of metric data.

FIG. 17 shows an architecture of an example metric data analytics system that may be implemented as part of a management system.

FIG. 18 shows an example implementation of an analytics services manager.

FIG. 19 shows an example of a history of metric data maintained by a metric processor of the forecast engine.

FIG. 20 shows an overview of example processing operations carried out by a metric processor.

FIGS. 21A-21C show an example of computing a trend estimate and detrending metric data within a historical window.

FIG. 22 shows example autoregressive and moving-average weight parameters for three autoregressive moving-average models.

FIG. 23 shows an example of a latest non-trendy metric data value and three forecasted non-trendy metric data values with the same time stamp.

FIG. 24 shows an example sequence of forecasted non-trendy metric data.

FIG. 25 shows a plot of an example stream of metric data that exhibits a pulse wave pattern.

FIG. 26 shows a plot of an example stream of metric data that exhibits a seasonal wave pattern.

FIGS. 27A-27B shows edge detection applied to a sequence of metric data.

FIG. 28 shows pulse widths and periods of the pulses in the pulse-wave stream of metric data of FIG. 25.

FIG. 29 shows a bar graph of four different examples of coarse sampling rates and associated streams of metric data.

FIG. 30 shows an example of periodograms for a series of short-time windows of non-trendy metric data.

FIG. 31A show a plot of a periodogram.

FIG. 31B shows a plot of an autocorrelation function that corresponds to the periodogram shown in FIG. 29A.

FIG. 31C shows examples of a local maximum and a local minimum in neighborhoods of the autocorrelation function shown in FIG. 29B.

FIG. 32A shows a plot of example periodic parameters for a pulse wave model.

FIG. 32B shows principle frequencies and an example histogram of principle frequencies.

FIG. 33A shows a plot of example trendy, non-periodic metric data and forecasted metric data over a forecast interval.

FIG. 33B shows a plot of example trendy, pulse-wave metric data and forecasted metric data over a forecast interval.

FIG. 33C shows a plot of example trendy, periodic metric data and forecasted metric data over a forecast interval.

FIGS. 34A-34D show example plots of confidence intervals.

FIGS. 35A-35B show example plots of a linearly increasing stochastic sequence of metric data and an increasing seasonally varying sequence of metric data, respectively.

FIGS. 36A-36B show example plots of a linearly increasing stochastic sequence of metric data and an increasing seasonally varying sequence of metric data, respectively.

FIG. 37 shows an example of incrementally increasing usable capacity of a resource.

FIG. 38 shows a control-flow diagram of a method to estimate time remaining and determine right-size for usable capacity of a resource of a distributed computing system.

FIG. 39 shows a control-flow diagram of a routine “remove trend from the stream” called in FIG. 38.

FIG. 40 shows a control-flow diagram of a routine “compute stochastic process models” called in FIG. 38.

FIG. 41 shows a control-flow diagram of a routine “compute periodic models” called in FIG. 38.

FIG. 42 shows a control-flow diagram of a routine “apply edge detection” called in FIG. 41.

FIG. 43 shows a control-flow diagram of a routine “compute seasonal period parameters” called in FIG. 41.

FIG. 44 shows control-flow diagram of a routine “compute period of stream” called in FIG. 43.

FIG. 45 shows a control-flow diagram a routine “compute forecast” called in FIG. 38.

FIG. 46 shows a control-flow diagram of the routine “estimate time remaining and right-size for usable capacity of resource” called in FIG. 38.

FIG. 47 shows a control-flow diagram of the routine “determine estimated time remaining for resource” called in FIG. 46.

FIG. 48 shows a control-flow diagram of the routine “determine right size for usable capacity of resource” called in FIG. 46.

DETAILED DESCRIPTION

Computational methods and systems for estimating time remaining and right sizing usable capacities of resources used to run virtual objects of a distributed computing system are described herein. In a first subsection, computer hardware, complex computational systems, and virtualization are described. Containers and containers supported by virtualization layers are described in a second subsection. Methods for estimating time remaining and right sizing usable capacities of resources are described below in a fourth subsection.

Computer Hardware, Complex Computational Systems, and Virtualization

The term “abstraction” is not, in any way, intended to mean or suggest an abstract idea or concept. Computational abstractions are tangible, physical interfaces that are implemented using physical computer hardware, data-storage devices, and communications systems. Instead, the term “abstraction” refers, in the current discussion, to a logical level of functionality encapsulated within one or more concrete, tangible, physically-implemented computer systems with defined interfaces through which electronically-encoded data is exchanged, process execution launched, and electronic services are provided. Interfaces may include graphical and textual data displayed on physical display devices as well as computer programs and routines that control physical computer processors to carry out various tasks and operations and that are invoked through electronically implemented application programming interfaces (“APIs”) and other electronically implemented interfaces. Software is essentially a sequence of encoded symbols, such as a printout of a computer program or digitally encoded computer instructions sequentially stored in a file on an optical disk or within an electromechanical mass-storage device. Software alone can do nothing. It is only when encoded computer instructions are loaded into an electronic memory within a computer system and executed on a physical processor that “software implemented” functionality is provided. The digitally encoded computer instructions are a physical control component of processor-controlled machines and devices. Multi-cloud aggregations, cloud-computing services, virtual-machine containers and virtual machines, containers, communications interfaces, and many of the other topics discussed below are tangible, physical components of physical, electro-optical-mechanical computer systems.

FIG. 1 shows a general architectural diagram for various types of computers. Computers that receive, process, and store event messages may be described by the general architectural diagram shown in FIG. 1, for example. The computer system contains one or multiple central processing units (“CPUs”) 102-105, one or more electronic memories 108 interconnected with the CPUs by a CPU/memory-subsystem bus 110 or multiple busses, a first bridge 112 that interconnects the CPU/memory-subsystem bus 110 with additional busses 114 and 116, or other types of high-speed interconnection media, including multiple, high-speed serial interconnects. These busses or serial interconnections, in turn, connect the CPUs and memory with specialized processors, such as a graphics processor 118, and with one or more additional bridges 120, which are interconnected with high-speed serial links or with multiple controllers 122-127, such as controller 127, that provide access to various different types of mass-storage devices 128, electronic displays, input devices, and other such components, subcomponents, and computational devices. It should be noted that computer-readable data-storage devices include optical and electromagnetic disks, electronic memories, and other physical data-storage devices. Those familiar with modern science and technology appreciate that electromagnetic radiation and propagating signals do not store data for subsequent retrieval, and can transiently “store” only a byte or less of information per mile, far less information than needed to encode even the simplest of routines.

Of course, there are many different types of computer-system architectures that differ from one another in the number of different memories, including different types of hierarchical cache memories, the number of processors and the connectivity of the processors with other system components, the number of internal communications busses and serial links, and in many other ways. However, computer systems generally execute stored programs by fetching instructions from memory and executing the instructions in one or more processors. Computer systems include general-purpose computer systems, such as personal computers (“PCs”), various types of server computers and workstations, and higher-end mainframe computers, but may also include a plethora of various types of special-purpose computing devices, including data-storage systems, communications routers, network nodes, tablet computers, and mobile telephones.

FIG. 2 shows an Internet-connected distributed computer system. As communications and networking technologies have evolved in capability and accessibility, and as the computational bandwidths, data-storage capacities, and other capabilities and capacities of various types of computer systems have steadily and rapidly increased, much of modern computing now generally involves large distributed systems and computers interconnected by local networks, wide-area networks, wireless communications, and the Internet. FIG. 2 shows a typical distributed system in which a large number of PCs 202-205, a high-end distributed mainframe system 210 with a large data-storage system 212, and a large computer center 214 with large numbers of rack-mounted server computers or blade servers all interconnected through various communications and networking systems that together comprise the Internet 216. Such distributed computing systems provide diverse arrays of functionalities. For example, a PC user may access hundreds of millions of different web sites provided by hundreds of thousands of different web servers throughout the world and may access high-computational-bandwidth computing services from remote computer facilities for running complex computational tasks.

Until recently, computational services were generally provided by computer systems and data centers purchased, configured, managed, and maintained by service-provider organizations. For example, an e-commerce retailer generally purchased, configured, managed, and maintained a data center including numerous web server computers, back-end computer systems, and data-storage systems for serving web pages to remote customers, receiving orders through the web-page interface, processing the orders, tracking completed orders, and other myriad different tasks associated with an e-commerce enterprise.

FIG. 3 shows cloud computing. In the recently developed cloud-computing paradigm, computing cycles and data-storage facilities are provided to organizations and individuals by cloud-computing providers. In addition, larger organizations may elect to establish private cloud-computing facilities in addition to, or instead of, subscribing to computing services provided by public cloud-computing service providers. In FIG. 3, a system administrator for an organization, using a PC 302, accesses the organization's private cloud 304 through a local network 306 and private-cloud interface 308 and also accesses, through the Internet 310, a public cloud 312 through a public-cloud services interface 314. The administrator can, in either the case of the private cloud 304 or public cloud 312, configure virtual computer systems and even entire virtual data centers and launch execution of application programs on the virtual computer systems and virtual data centers in order to carry out any of many different types of computational tasks. As one example, a small organization may configure and run a virtual data center within a public cloud that executes web servers to provide an e-commerce interface through the public cloud to remote customers of the organization, such as a user viewing the organization's e-commerce web pages on a remote user system 316.

Cloud-computing facilities are intended to provide computational bandwidth and data-storage services much as utility companies provide electrical power and water to consumers. Cloud computing provides enormous advantages to small organizations without the devices to purchase, manage, and maintain in-house data centers. Such organizations can dynamically add and delete virtual computer systems from their virtual data centers within public clouds in order to track computational-bandwidth and data-storage needs, rather than purchasing sufficient computer systems within a physical data center to handle peak computational-bandwidth and data-storage demands. Moreover, small organizations can completely avoid the overhead of maintaining and managing physical computer systems, including hiring and periodically retraining information-technology specialists and continuously paying for operating-system and database-management-system upgrades. Furthermore, cloud-computing interfaces allow for easy and straightforward configuration of virtual computing facilities, flexibility in the types of applications and operating systems that can be configured, and other functionalities that are useful even for owners and administrators of private cloud-computing facilities used by a single organization.

FIG. 4 shows generalized hardware and software components of a general-purpose computer system, such as a general-purpose computer system having an architecture similar to that shown in FIG. 1. The computer system 400 is often considered to include three fundamental layers: (1) a hardware layer or level 402; (2) an operating-system layer or level 404; and (3) an application-program layer or level 406. The hardware layer 402 includes one or more processors 408, system memory 410, various different types of input-output (“I/O”) devices 410 and 412, and mass-storage devices 414. Of course, the hardware level also includes many other components, including power supplies, internal communications links and busses, specialized integrated circuits, many different types of processor-controlled or microprocessor-controlled peripheral devices and controllers, and many other components. The operating system 404 interfaces to the hardware level 402 through a low-level operating system and hardware interface 416 generally comprising a set of non-privileged computer instructions 418, a set of privileged computer instructions 420, a set of non-privileged registers and memory addresses 422, and a set of privileged registers and memory addresses 424. In general, the operating system exposes non-privileged instructions, non-privileged registers, and non-privileged memory addresses 426 and a system-call interface 428 as an operating-system interface 430 to application programs 432-436 that execute within an execution environment provided to the application programs by the operating system. The operating system, alone, accesses the privileged instructions, privileged registers, and privileged memory addresses. By reserving access to privileged instructions, privileged registers, and privileged memory addresses, the operating system can ensure that application programs and other higher-level computational entities cannot interfere with one another's execution and cannot change the overall state of the computer system in ways that could deleteriously impact system operation. The operating system includes many internal components and modules, including a scheduler 442, memory management 444, a file system 446, device drivers 448, and many other components and modules. To a certain degree, modern operating systems provide numerous levels of abstraction above the hardware level, including virtual memory, which provides to each application program and other computational entities a separate, large, linear memory-address space that is mapped by the operating system to various electronic memories and mass-storage devices. The scheduler orchestrates interleaved execution of various different application programs and higher-level computational entities, providing to each application program a virtual, stand-alone system devoted entirely to the application program. From the application program's standpoint, the application program executes continuously without concern for the need to share processor devices and other system devices with other application programs and higher-level computational entities. The device drivers abstract details of hardware-component operation, allowing application programs to employ the system-call interface for transmitting and receiving data to and from communications networks, mass-storage devices, and other I/O devices and subsystems. The file system 446 facilitates abstraction of mass-storage-device and memory devices as a high-level, easy-to-access, file-system interface. Thus, the development and evolution of the operating system has resulted in the generation of a type of multi-faceted virtual execution environment for application programs and other higher-level computational entities.

While the execution environments provided by operating systems have proved to be an enormously successful level of abstraction within computer systems, the operating-system-provided level of abstraction is nonetheless associated with difficulties and challenges for developers and users of application programs and other higher-level computational entities. One difficulty arises from the fact that there are many different operating systems that run within various different types of computer hardware. In many cases, popular application programs and computational systems are developed to run on only a subset of the available operating systems and can therefore be executed within only a subset of the different types of computer systems on which the operating systems are designed to run. Often, even when an application program or other computational system is ported to additional operating systems, the application program or other computational system can nonetheless run more efficiently on the operating systems for which the application program or other computational system was originally targeted. Another difficulty arises from the increasingly distributed nature of computer systems. Although distributed operating systems are the subject of considerable research and development efforts, many of the popular operating systems are designed primarily for execution on a single computer system. In many cases, it is difficult to move application programs, in real time, between the different computer systems of a distributed computer system for high-availability, fault-tolerance, and load-balancing purposes. The problems are even greater in heterogeneous distributed computer systems which include different types of hardware and devices running different types of operating systems. Operating systems continue to evolve, as a result of which certain older application programs and other computational entities may be incompatible with more recent versions of operating systems for which they are targeted, creating compatibility issues that are particularly difficult to manage in large distributed systems.

For all of these reasons, a higher level of abstraction, referred to as the “virtual machine,” (“VM”) has been developed and evolved to further abstract computer hardware in order to address many difficulties and challenges associated with traditional computing systems, including the compatibility issues discussed above. FIGS. 5A-B show two types of VM and virtual-machine execution environments. FIGS. 5A-B use the same illustration conventions as used in FIG. 4. FIG. 5A shows a first type of virtualization. The computer system 500 in FIG. 5A includes the same hardware layer 502 as the hardware layer 402 shown in FIG. 4. However, rather than providing an operating system layer directly above the hardware layer, as in FIG. 4, the virtualized computing environment shown in FIG. 5A features a virtualization layer 504 that interfaces through a virtualization-layer/hardware-layer interface 506, equivalent to interface 416 in FIG. 4, to the hardware. The virtualization layer 504 provides a hardware-like interface to a number of VMs, such as VM 510, in a virtual-machine layer 511 executing above the virtualization layer 504. Each VM includes one or more application programs or other higher-level computational entities packaged together with an operating system, referred to as a “guest operating system,” such as application 514 and guest operating system 516 packaged together within VM 510. Each VM is thus equivalent to the operating-system layer 404 and application-program layer 406 in the general-purpose computer system shown in FIG. 4. Each guest operating system within a VM interfaces to the virtualization layer interface 504 rather than to the actual hardware interface 506. The virtualization layer 504 partitions hardware devices into abstract virtual-hardware layers to which each guest operating system within a VM interfaces. The guest operating systems within the VMs, in general, are unaware of the virtualization layer and operate as if they were directly accessing a true hardware interface. The virtualization layer 504 ensures that each of the VMs currently executing within the virtual environment receive a fair allocation of underlying hardware devices and that all VMs receive sufficient devices to progress in execution. The virtualization layer 504 may differ for different guest operating systems. For example, the virtualization layer is generally able to provide virtual hardware interfaces for a variety of different types of computer hardware. This allows, as one example, a VM that includes a guest operating system designed for a particular computer architecture to run on hardware of a different architecture. The number of VMs need not be equal to the number of physical processors or even a multiple of the number of processors.

The virtualization layer 504 includes a virtual-machine-monitor module 518 (“VMM”) that virtualizes physical processors in the hardware layer to create virtual processors on which each of the VMs executes. For execution efficiency, the virtualization layer attempts to allow VMs to directly execute non-privileged instructions and to directly access non-privileged registers and memory. However, when the guest operating system within a VM accesses virtual privileged instructions, virtual privileged registers, and virtual privileged memory through the virtualization layer 504, the accesses result in execution of virtualization-layer code to simulate or emulate the privileged devices. The virtualization layer additionally includes a kernel module 520 that manages memory, communications, and data-storage machine devices on behalf of executing VMs (“VM kernel”). The VM kernel, for example, maintains shadow page tables on each VM so that hardware-level virtual-memory facilities can be used to process memory accesses. The VM kernel additionally includes routines that implement virtual communications and data-storage devices as well as device drivers that directly control the operation of underlying hardware communications and data-storage devices. Similarly, the VM kernel virtualizes various other types of I/O devices, including keyboards, optical-disk drives, and other such devices. The virtualization layer 504 essentially schedules execution of VMs much like an operating system schedules execution of application programs, so that the VMs each execute within a complete and fully functional virtual hardware layer.

FIG. 5B shows a second type of virtualization. In FIG. 5B, the computer system 540 includes the same hardware layer 542 and operating system layer 544 as the hardware layer 402 and the operating system layer 404 shown in FIG. 4. Several application programs 546 and 548 are shown running in the execution environment provided by the operating system 544. In addition, a virtualization layer 550 is also provided, in computer 540, but, unlike the virtualization layer 504 discussed with reference to FIG. 5A, virtualization layer 550 is layered above the operating system 544, referred to as the “host OS,” and uses the operating system interface to access operating-system-provided functionality as well as the hardware. The virtualization layer 550 comprises primarily a VMM and a hardware-like interface 552, similar to hardware-like interface 508 in FIG. 5A. The hardware-layer interface 552, equivalent to interface 416 in FIG. 4, provides an execution environment for a number of VMs 556-558, each including one or more application programs or other higher-level computational entities packaged together with a guest operating system.

In FIGS. 5A-5B, the layers are somewhat simplified for clarity of illustration. For example, portions of the virtualization layer 550 may reside within the host-operating-system kernel, such as a specialized driver incorporated into the host operating system to facilitate hardware access by the virtualization layer.

It should be noted that virtual hardware layers, virtualization layers, and guest operating systems are all physical entities that are implemented by computer instructions stored in physical data-storage devices, including electronic memories, mass-storage devices, optical disks, magnetic disks, and other such devices. The term “virtual” does not, in any way, imply that virtual hardware layers, virtualization layers, and guest operating systems are abstract or intangible. Virtual hardware layers, virtualization layers, and guest operating systems execute on physical processors of physical computer systems and control operation of the physical computer systems, including operations that alter the physical states of physical devices, including electronic memories and mass-storage devices. They are as physical and tangible as any other component of a computer since, such as power supplies, controllers, processors, busses, and data-storage devices.

A VM or virtual application, described below, is encapsulated within a data package for transmission, distribution, and loading into a virtual-execution environment. One public standard for virtual-machine encapsulation is referred to as the “open virtualization format” (“OVF”). The OVF standard specifies a format for digitally encoding a VM within one or more data files. FIG. 6 shows an OVF package. An OVF package 602 includes an OVF descriptor 604, an OVF manifest 606, an OVF certificate 608, one or more disk-image files 610-611, and one or more device files 612-614. The OVF package can be encoded and stored as a single file or as a set of files. The OVF descriptor 604 is an XML document 620 that includes a hierarchical set of elements, each demarcated by a beginning tag and an ending tag. The outermost, or highest-level, element is the envelope element, demarcated by tags 622 and 623. The next-level element includes a reference element 626 that includes references to all files that are part of the OVF package, a disk section 628 that contains meta information about all of the virtual disks included in the OVF package, a network section 630 that includes meta information about all of the logical networks included in the OVF package, and a collection of virtual-machine configurations 632 which further includes hardware descriptions of each VM 634. There are many additional hierarchical levels and elements within a typical OVF descriptor. The OVF descriptor is thus a self-describing, XML file that describes the contents of an OVF package. The OVF manifest 606 is a list of cryptographic-hash-function-generated digests 636 of the entire OVF package and of the various components of the OVF package. The OVF certificate 608 is an authentication certificate 640 that includes a digest of the manifest and that is cryptographically signed. Disk image files, such as disk image file 610, are digital encodings of the contents of virtual disks and device files 612 are digitally encoded content, such as operating-system images. A VM or a collection of VMs encapsulated together within a virtual application can thus be digitally encoded as one or more files within an OVF package that can be transmitted, distributed, and loaded using well-known tools for transmitting, distributing, and loading files. A virtual appliance is a software service that is delivered as a complete software stack installed within one or more VMs that is encoded within an OVF package.

The advent of VMs and virtual environments has alleviated many of the difficulties and challenges associated with traditional general-purpose computing. Machine and operating-system dependencies can be significantly reduced or entirely eliminated by packaging applications and operating systems together as VMs and virtual appliances that execute within virtual environments provided by virtualization layers running on many different types of computer hardware. A next level of abstraction, referred to as virtual data centers or virtual infrastructure, provide a data-center interface to virtual data centers computationally constructed within physical data centers.

FIG. 7 shows virtual data centers provided as an abstraction of underlying physical-data-center hardware components. In FIG. 7, a physical data center 702 is shown below a virtual-interface plane 704. The physical data center consists of a virtual-data-center management server computer 706 and any of various different computers, such as PC 708, on which a virtual-data-center management interface may be displayed to system administrators and other users. The physical data center additionally includes generally large numbers of server computers, such as server computer 710, that are coupled together by local area networks, such as local area network 712 that directly interconnects server computer 710 and 714-720 and a mass-storage array 722. The physical data center shown in FIG. 7 includes three local area networks 712, 724, and 726 that each directly interconnects a bank of eight server computers and a mass-storage array. The individual server computers, such as server computer 710, each includes a virtualization layer and runs multiple VMs. Different physical data centers may include many different types of computers, networks, data-storage systems and devices connected according to many different types of connection topologies. The virtual-interface plane 704, a logical abstraction layer shown by a plane in FIG. 7, abstracts the physical data center to a virtual data center comprising one or more device pools, such as device pools 730-732, one or more virtual data stores, such as virtual data stores 734-736, and one or more virtual networks. In certain implementations, the device pools abstract banks of server computers directly interconnected by a local area network.

The virtual-data-center management interface allows provisioning and launching of VMs with respect to device pools, virtual data stores, and virtual networks, so that virtual-data-center administrators need not be concerned with the identities of physical-data-center components used to execute particular VMs. Furthermore, the virtual-data-center management server computer 706 includes functionality to migrate running VMs from one server computer to another in order to optimally or near optimally manage device allocation, provides fault tolerance, and high availability by migrating VMs to most effectively utilize underlying physical hardware devices, to replace VMs disabled by physical hardware problems and failures, and to ensure that multiple VMs supporting a high-availability virtual appliance are executing on multiple physical computer systems so that the services provided by the virtual appliance are continuously accessible, even when one of the multiple virtual appliances becomes compute bound, data-access bound, suspends execution, or fails. Thus, the virtual data center layer of abstraction provides a virtual-data-center abstraction of physical data centers to simplify provisioning, launching, and maintenance of VMs and virtual appliances as well as to provide high-level, distributed functionalities that involve pooling the devices of individual server computers and migrating VMs among server computers to achieve load balancing, fault tolerance, and high availability.

FIG. 8 shows virtual-machine components of a virtual-data-center management server computer and physical server computers of a physical data center above which a virtual-data-center interface is provided by the virtual-data-center management server computer. The virtual-data-center management server computer 802 and a virtual-data-center database 804 comprise the physical components of the management component of the virtual data center. The virtual-data-center management server computer 802 includes a hardware layer 806 and virtualization layer 808, and runs a virtual-data-center management-server VM 810 above the virtualization layer. Although shown as a single server computer in FIG. 8, the virtual-data-center management server computer (“VDC management server”) may include two or more physical server computers that support multiple VDC-management-server virtual appliances. The virtual-data-center management-server VM 810 includes a management-interface component 812, distributed services 814, core services 816, and a host-management interface 818. The host-management interface 818 is accessed from any of various computers, such as the PC 708 shown in FIG. 7. The host-management interface 818 allows the virtual-data-center administrator to configure a virtual data center, provision VMs, collect statistics and view log files for the virtual data center, and to carry out other, similar management tasks. The host-management interface 818 interfaces to virtual-data-center agents 824, 825, and 826 that execute as VMs within each of the server computers of the physical data center that is abstracted to a virtual data center by the VDC management server computer.

The distributed services 814 include a distributed-device scheduler that assigns VMs to execute within particular physical server computers and that migrates VMs in order to most effectively make use of computational bandwidths, data-storage capacities, and network capacities of the physical data center. The distributed services 814 further include a high-availability service that replicates and migrates VMs in order to ensure that VMs continue to execute despite problems and failures experienced by physical hardware components. The distributed services 814 also include a live-virtual-machine migration service that temporarily halts execution of a VM, encapsulates the VM in an OVF package, transmits the OVF package to a different physical server computer, and restarts the VM on the different physical server computer from a virtual-machine state recorded when execution of the VM was halted. The distributed services 814 also include a distributed backup service that provides centralized virtual-machine backup and restore.

The core services 816 provided by the VDC management server VM 810 include host configuration, virtual-machine configuration, virtual-machine provisioning, generation of virtual-data-center alerts and events, ongoing event logging and statistics collection, a task scheduler, and a device-management module. Each physical server computers 820-822 also includes a host-agent VM 828-830 through which the virtualization layer can be accessed via a virtual-infrastructure application programming interface (“API”). This interface allows a remote administrator or user to manage an individual server computer through the infrastructure API. The virtual-data-center agents 824-826 access virtualization-layer server information through the host agents. The virtual-data-center agents are primarily responsible for offloading certain of the virtual-data-center management-server functions specific to a particular physical server to that physical server computer. The virtual-data-center agents relay and enforce device allocations made by the VDC management server VM 810, relay virtual-machine provisioning and configuration-change commands to host agents, monitor and collect performance statistics, alerts, and events communicated to the virtual-data-center agents by the local host agents through the interface API, and to carry out other, similar virtual-data-management tasks.

The virtual-data-center abstraction provides a convenient and efficient level of abstraction for exposing the computational devices of a cloud-computing facility to cloud-computing-infrastructure users. A cloud-director management server exposes virtual devices of a cloud-computing facility to cloud-computing-infrastructure users. In addition, the cloud director introduces a multi-tenancy layer of abstraction, which partitions VDCs into tenant-associated VDCs that can each be allocated to a particular individual tenant or tenant organization, both referred to as a “tenant.” A given tenant can be provided one or more tenant-associated VDCs by a cloud director managing the multi-tenancy layer of abstraction within a cloud-computing facility. The cloud services interface (308 in FIG. 3) exposes a virtual-data-center management interface that abstracts the physical data center.

FIG. 9 shows a cloud-director level of abstraction. In FIG. 9, three different physical data centers 902-904 are shown below planes representing the cloud-director layer of abstraction 906-908. Above the planes representing the cloud-director level of abstraction, multi-tenant virtual data centers 910-912 are shown. The devices of these multi-tenant virtual data centers are securely partitioned in order to provide secure virtual data centers to multiple tenants, or cloud-services-accessing organizations. For example, a cloud-services-provider virtual data center 910 is partitioned into four different tenant-associated virtual-data centers within a multi-tenant virtual data center for four different tenants 916-919. Each multi-tenant virtual data center is managed by a cloud director comprising one or more cloud-director server computers 920-922 and associated cloud-director databases 924-926. Each cloud-director server computer or server computers runs a cloud-director virtual appliance 930 that includes a cloud-director management interface 932, a set of cloud-director services 934, and a virtual-data-center management-server interface 936. The cloud-director services include an interface and tools for provisioning multi-tenant virtual data center virtual data centers on behalf of tenants, tools and interfaces for configuring and managing tenant organizations, tools and services for organization of virtual data centers and tenant-associated virtual data centers within the multi-tenant virtual data center, services associated with template and media catalogs, and provisioning of virtualization networks from a network pool. Templates are VMs that each contains an OS and/or one or more VMs containing applications. A template may include much of the detailed contents of VMs and virtual appliances that are encoded within OVF packages, so that the task of configuring a VM or virtual appliance is significantly simplified, requiring only deployment of one OVF package. These templates are stored in catalogs within a tenant's virtual-data center. These catalogs are used for developing and staging new virtual appliances and published catalogs are used for sharing templates in virtual appliances across organizations. Catalogs may include OS images and other information relevant to construction, distribution, and provisioning of virtual appliances.

Considering FIGS. 7 and 9, the VDC-server and cloud-director layers of abstraction can be seen, as discussed above, to facilitate employment of the virtual-data-center concept within private and public clouds. However, this level of abstraction does not fully facilitate aggregation of single-tenant and multi-tenant virtual data centers into heterogeneous or homogeneous aggregations of cloud-computing facilities.

FIG. 10 shows virtual-cloud-connector nodes (“VCC nodes”) and a VCC server, components of a distributed system that provides multi-cloud aggregation and that includes a cloud-connector server and cloud-connector nodes that cooperate to provide services that are distributed across multiple clouds. VMware vCloud™ VCC servers and nodes are one example of VCC server and nodes. In FIG. 10, seven different cloud-computing facilities are shown 1002-1008. Cloud-computing facility 1002 is a private multi-tenant cloud with a cloud director 1010 that interfaces to a VDC management server 1012 to provide a multi-tenant private cloud comprising multiple tenant-associated virtual data centers. The remaining cloud-computing facilities 1003-1008 may be either public or private cloud-computing facilities and may be single-tenant virtual data centers, such as virtual data centers 1003 and 1006, multi-tenant virtual data centers, such as multi-tenant virtual data centers 1004 and 1007-1008, or any of various different kinds of third-party cloud-services facilities, such as third-party cloud-services facility 1005. An additional component, the VCC server 1014, acting as a controller is included in the private cloud-computing facility 1002 and interfaces to a VCC node 1016 that runs as a virtual appliance within the cloud director 1010. A VCC server may also run as a virtual appliance within a VDC management server that manages a single-tenant private cloud. The VCC server 1014 additionally interfaces, through the Internet, to VCC node virtual appliances executing within remote VDC management servers, remote cloud directors, or within the third-party cloud services 1018-1023. The VCC server provides a VCC server interface that can be displayed on a local or remote terminal, PC, or other computer system 1026 to allow a cloud-aggregation administrator or other user to access VCC-server-provided aggregate-cloud distributed services. In general, the cloud-computing facilities that together form a multiple-cloud-computing aggregation through distributed services provided by the VCC server and VCC nodes are geographically and operationally distinct.

Containers and Containers Supported by Virtualization Layers

As mentioned above, while the virtual-machine-based virtualization layers, described in the previous subsection, have received widespread adoption and use in a variety of different environments, from personal computers to enormous distributed computing systems, traditional virtualization technologies are associated with computational overheads. While these computational overheads have steadily decreased, over the years, and often represent ten percent or less of the total computational bandwidth consumed by an application running above a guest operating system in a virtualized environment, traditional virtualization technologies nonetheless involve computational costs in return for the power and flexibility that they provide.

While a traditional virtualization layer can simulate the hardware interface expected by any of many different operating systems, OSL virtualization essentially provides a secure partition of the execution environment provided by a particular operating system. As one example, OSL virtualization provides a file system to each container, but the file system provided to the container is essentially a view of a partition of the general file system provided by the underlying operating system of the host. In essence, OSL virtualization uses operating-system features, such as namespace isolation, to isolate each container from the other containers running on the same host. In other words, namespace isolation ensures that each application is executed within the execution environment provided by a container to be isolated from applications executing within the execution environments provided by the other containers. A container cannot access files not included the container's namespace and cannot interact with applications running in other containers. As a result, a container can be booted up much faster than a VM, because the container uses operating-system-kernel features that are already available and functioning within the host. Furthermore, the containers share computational bandwidth, memory, network bandwidth, and other computational resources provided by the operating system, without the overhead associated with computational resources allocated to VMs and virtualization layers. Again, however, OSL virtualization does not provide many desirable features of traditional virtualization. As mentioned above, OSL virtualization does not provide a way to run different types of operating systems for different groups of containers within the same host and OSL-virtualization does not provide for live migration of containers between hosts, high-availability functionality, distributed resource scheduling, and other computational functionality provided by traditional virtualization technologies.

FIG. 11 shows an example server computer used to host three containers. As discussed above with reference to FIG. 4, an operating system layer 404 runs above the hardware 402 of the host computer. The operating system provides an interface, for higher-level computational entities, that includes a system-call interface 428 and the non-privileged instructions, memory addresses, and registers 426 provided by the hardware layer 402. However, unlike in FIG. 4, in which applications run directly above the operating system layer 404, OSL virtualization involves an OSL virtualization layer 1102 that provides operating-system interfaces 1104-1106 to each of the containers 1108-1110. The containers, in turn, provide an execution environment for an application that runs within the execution environment provided by container 1108. The container can be thought of as a partition of the resources generally available to higher-level computational entities through the operating system interface 430.

FIG. 12 shows an approach to implementing the containers on a VM. FIG. 12 shows a host computer similar to that shown in FIG. 5A, discussed above. The host computer includes a hardware layer 502 and a virtualization layer 504 that provides a virtual hardware interface 508 to a guest operating system 1102. Unlike in FIG. 5A, the guest operating system interfaces to an OSL-virtualization layer 1104 that provides container execution environments 1206-1208 to multiple application programs.

Note that, although only a single guest operating system and OSL virtualization layer are shown in FIG. 12, a single virtualized host system can run multiple different guest operating systems within multiple VMs, each of which supports one or more OSL-virtualization containers. A virtualized, distributed computing system that uses guest operating systems running within VMs to support OSL-virtualization layers to provide containers for running applications is referred to, in the following discussion, as a “hybrid virtualized distributed computing system.”

Running containers above a guest operating system within a VM provides advantages of traditional virtualization in addition to the advantages of OSL virtualization. Containers can be quickly booted in order to provide additional execution environments and associated resources for additional application instances. The resources available to the guest operating system are efficiently partitioned among the containers provided by the OSL-virtualization layer 1204 in FIG. 12, because there is almost no additional computational overhead associated with container-based partitioning of computational resources. However, many of the powerful and flexible features of the traditional virtualization technology can be applied to VMs in which containers run above guest operating systems, including live migration from one host to another, various types of high-availability and distributed resource scheduling, and other such features. Containers provide share-based allocation of computational resources to groups of applications with guaranteed isolation of applications in one container from applications in the remaining containers executing above a guest operating system. Moreover, resource allocation can be modified at run time between containers. The traditional virtualization layer provides for flexible and scaling over large numbers of hosts within large distributed computing systems and a simple approach to operating-system upgrades and patches. Thus, the use of OSL virtualization above traditional virtualization in a hybrid virtualized distributed computing system, as shown in FIG. 12, provides many of the advantages of both a traditional virtualization layer and the advantages of OSL virtualization.

Method and System for Estimating Time Remaining and Right Sizing Usable Capacities of Resources Used to Run Virtual Objects of a Distributed Computing System

FIG. 13 shows an example of a virtualization layer 1302 located above a physical data center 1304. The virtualization layer 1302 is separated from the physical data center 1304 by a virtual-interface plane 1306. The physical data center 1304 comprises a management server computer 1308 and any of various computers, such as PC 1310, on which a virtual-data-center management interface may be displayed to system administrators and other users. The physical data center 1304 additionally includes many server computers, such as server computers 1312-1319, that are coupled together by local area networks 1320-1322. In the example of FIG. 13, each local area network directly interconnects a bank of eight server computers and a mass-storage array. For example, local area network 1320 directly interconnects server computers 1312-1319 and a mass-storage array 1324. Different physical data centers may be composed of many different types of computers, networks, data-storage systems and devices connected according to many different types of connection topologies. In the example of FIG. 13, the virtualization layer 1302 includes six virtual objects denoted by VO₁, VO₇, VO₃, VO₄, VO₅, and VO₆. A virtual object can be an application, a VM, or a container. The virtual objects VO₁ and VO₂ are hosted by a server computer 1326. The virtual objects VO₃, VO₄, VO₅, and VO₆ are hosted by a cluster of server computers 1313-1315. The virtualization layer 1302 includes virtual data stores 1332 and 1334 that provide virtual storage for the virtual objects. FIG. 13 also shows a virtual infrastructure management tool (“VIMT”) 1336 abstracted to the virtualization layer 1302. The VIMT 1336 is hosted by the management server computer 1308. The VIMT 1336 includes an information technology (“IT”) operations management server, such as VMware's vRealize® Operations™.

In the following discussion, the term “resource” means an actual physical resource of a distributed computing system, such as a multicore processor, a core, memory, disk drive, network adapter, or a storage appliance. Resources of a server computer and clusters of server computers form a resource pool for creating virtual resources of a virtual infrastructure used to run virtual objects. The term “virtual resource” refers to a portion of one or more resources assigned to a virtual object. For example, a virtual resource may be one or more cores of a multicore processor, a portion of memory, or a sector of a disk drive for use by a virtual object. Each virtual object uses only the resources of the resource pool assigned to the virtual object.

FIG. 14A shows an example of a virtual infrastructure formed from resources of the server computer 1336 used to run the two virtual objects VO₁ and VO₂ shown in FIG. 13. In this example, the resources of the server computer 1336 include a quad core processor 1402 with four independent processor cores, memory 1404, disk drive 1406, and a network adaptor 1408. The resources of the server computer 1336 are a resource pool that is partitioned to form a virtual infrastructure used to run the virtual objects VO₁ and VO₂. For example, two cores 1410 and 1412 of the quad core processor 1402 correspond to two virtual processors 1414 and 1416 assigned to the virtual object VO₁. Virtual storage 1417 comprises a portion of the disk drive 1406 assigned to the virtual object VO₁. The network adaptor 1408 serves as a virtual adaptor 1418 that provides communication between the virtual objects VO₁ and VO₂, between the virtual object VO₁ and other virtual objects on different hosts, and between the virtual object VO₁ and physical machines of the distributed computer system. Virtual memory 1420 comprises a portion of the memory 1404 assigned to the virtual object VO₁. In a similar manner, virtual processors 1422 and 1424, virtual storage 1426, virtual network adaptor 1428, and virtual memory 1430 are assigned to the virtual object VO₂ from the resources of the server computer 1336. The virtual processors 1414, 1416, 1422, and 1424, virtual storage 1417 and 1426, virtual network adaptors 1418 and 1428, and virtual memories 1420 and 1428 are virtual resources of a virtual infrastructure created from the resources of the server computer 1336.

A resource pool is not limited to the resources of a single server computer. A resource pool may be composed of the resources of a cluster of server computers that in turn are partitioned to from a virtual infrastructure used to run multiple virtual objects.

FIG. 14B shows an example of a virtual infrastructure formed from resources of the cluster of server computers 1313-1315 used to run the four virtual objects VO₃, VO₄, VO₅, and VO₆ shown in FIG. 13. In this example, the resources of the cluster of server computers 1313-1315 include quad core processors 1432-1434, memories 1436-1438, disk drives 1440-1442, and network adaptors 1444-1446. The resources of the cluster of server computers 1313-1315 are a resource pool partitioned to form a virtual infrastructure 1448 to run the virtual objects VO₃, VO₄, VO₅, and VO₆. For example, two cores 1450 and 1451 of the processor 1432 and core 1452 of the processor 1433 are three virtual processors 1454-1456 assigned to the virtual object VO₃. Virtual storage 1458 comprises a portion of the physical disk drives 1440 and 1441 assigned to the virtual object VO₃. The network adaptor 1444 serves as a virtual network adaptor 1460 that provides communication between the virtual object VO₃ and the virtual objects VO₄, VO₅, and VO₆ and between the virtual object VO₃ and server computers and devices of the distributed computer system. Virtual memory 1462 comprises a portion of the memory 1436 and memory 1437 assigned to the virtual object VO₃. In a similar manner, the resources of the server computers 1313-1315 are used to create the virtual processors 1464-1466, virtual storage 1468-1470, virtual network adaptors 1472-1474, and virtual memories 1476-1478 of the virtual infrastructure 1448 are assigned to the virtual objects VO₄, VO₅, and VO₆.

Workloads of virtual objects often vary with time, leading to wasted resources and unnecessary costs to tenants. For example, a tenant of a distributed computing system may have created numerous additional VMs to meet a temporary increase in demand for services provided by the tenant. But when the demand has returned to normal the tenant may have forgotten to delete the additional VMs, in which case the VMs become idle. Alternatively, the tenant may have powered off the additional virtual objects believing the additional VMs will be needed in the near future. In the following discussion, idle or powered off virtual objects are called “inactive virtual objects.” The resources allocated to inactive virtual objects are wasted, because these resources cannot be utilized by other virtual objects while the resources remain assigned to the inactive virtual objects. The cloud computing service provider is not able to maximize resource usage and the tenant may unknowingly continue to pay for use of resources that are assigned to the inactive virtual objects.

The VIMT 1336 manages virtual resources of a virtual infrastructure formed from resources that are assigned to the virtual objects. The VIMT 1336 monitors usage of virtual resources in the virtual infrastructure based on streams of time series metric data, also called “streams of metric data” or “metric data streams,” sent to the VIMT 1336 from operating systems, guest operating systems, and other metric data sources running on the server computers, computer systems, network devices, and mass-storage devices. As described below, the VIMT 1336 manages virtual resources of a virtual infrastructure by forecasting usage of virtual resources by virtual objects, identifying virtual objects that are expected to be inactive in the future based on the forecasted usage of virtual resources, and reclaiming the virtual resources assigned to the prospective inactive virtual objects for use by active virtual objects.

FIG. 15 shows the VIMT 1336 receiving streams of metric data represented by directional arrows 1502-1506. The streams of metric data include core usage, memory usage, network throughput, network traffic, amount of storage in use, hard drive active time, and hard disk transfer rate. Core usage is a measure of processor time used to process instructions of an application program or operating system as a percentage of core capacity. Amount of memory is the amount of memory (e.g., GBs) a computer system or other device uses at a given time. Network throughput is the number of bits of data transmitted to and from a server computer or data-storage device per unit of time and is often recorded in megabits, kilobits or simply bits per second. Network traffic at a server computer or mass-storage array is a count of the number of data packets received and sent at a given time. Clusters of server computers may also send collective metric data to the VIMT 1336. For example, a cluster of server computers 1312-1319 sends streams of cluster metric data, such as total processor usage, total amount of memory usage, total network throughput, total network traffic, disk active time, hard disk transfer rate to the VIMT 1336.

FIG. 16A-16D show plots of four different example streams of metric data. Horizontal axes, such as axis 1602, represents time. Vertical axes, such as vertical axis 1604, represents a range of metric data amplitudes. In FIGS. 16A-16C, curves represent four examples of different patterns of metric data streams. For example, in FIG. 16A, curve 1606 represents a periodic stream of metric data in which the pattern of metric data in time interval 1608 is repeated. In FIG. 16B, curve 1610 represents a trendy stream of metric data in which the amplitude of the metric data generally increases with increasing time. In FIG. 16C, curve 1612 represents a non-trendy, non-periodic stream of metric data. In FIG. 16D, rectangles 1614-1617 represent pulse waves of a pulsed stream of metric data generated by a resource that is utilized periodically and only for the duration of each pulse. The example streams of time series metric data shown in FIGS. 16A-16D represent usage of different resources. For example, the metric data in FIG. 16A may represent CPU usage of a core in a multicore processor of a server computer over time. The metric data in FIG. 16B may represent the amount of virtual memory a VM uses over time. The metric data in FIG. 16C may represent network throughput for a cluster of server computers.

In FIGS. 16A-16D, the streams of metric data are represented by continuous curves. In practice, a stream of metric data comprises a sequence of discrete metric data values in which each numerical value is recorded in a data-storage device with a time stamp. FIG. 16A includes a magnified view 1618 of three consecutive metric data points represented by points. Points represent amplitudes of metric data points at corresponding time stamps. For example, points 1620-1622 represents consecutive metric data values (i.e., amplitudes) z_(k−1), z_(k), and z_(k+1) recorded in a data-storage device at corresponding time stamps t_(k−1), t_(k), and t_(k+1), where subscript k is an integer time index of the k-th metric data point in the stream of metric data.

FIG. 17 shows an architecture of an example metric data analytics system 1700 that may be implemented as part of the VIMT 1336. The analytics system 1700 comprises an analytics services manager 1702, a forecast engine 1704, and a metric data stream database 1706. The analytics services manager 1702 receives streams of metric data represented by directional arrows, such as directional arrow 1708. The forecast engine 1702 host a collection of metric processors, such as metric processors 1710-1712. The forecast engine 1704 provides a library of configurable models. The forecast engine 1704 includes an interface that enables a user to create one or more metric processors from the configurable models described below and assigns to each metric processor a single stream of metric data. Each metric processor is registered with a registration key that the analytical services manager 1702 uses to route a stream of metric data associate with a physical resource to a corresponding metric processor. Each stream of metric data is copied to the database 1706 to create a history for each resource. Each metric processor generates a forecast when the metric processor receives a forecast request from the VIMT 1336. The VIMT 1336 includes a workload placement application 1714, a capacity planning application 1716, or any other application 1718 that uses forecasted metric data. The VIMT 1336 may submit a request to the analytics services manager 1702 for forecasted active status metric data for a forecast interval to prospectively determine whether an associated virtual object is active or inactive. In the following discussion, the terms “stream of metric data,” “metric data stream,” “sequence of metric data,” and “forecasted metric data” includes “stream of active status metric data,” “active status metric data stream,” “sequence of active status metric data,” and “forecasted active status metric data.”

FIG. 18 shows an example implementation of the analytics services manager 1702. Each metric processor is registered with a resource key in the analytics services manager 1702. Each data point of a stream of metric data comprises a resource key, time stamp, and a metric data value. The analytics services manager 1702 utilizes the resource key to route the stream of metric data to the metric processor associated with the resource key. In the example of FIG. 18, a series of decision blocks 1801-1803 represent operations in which the resource key of each stream of metric data received by the analytics services manager 1702 is checked against the resource keys of registered metric processors. Blocks 1804-1806 represent forwarding operations that correspond to the decision blocks 1801-1803 in which a metric data stream with a resource key that matches one of the registered registration keys is forwarded to one of the corresponding metric processors 1710-1712 of the forecast engine 1704. For example, FIG. 18 shows an example stream of metric data 1808 with a resource key denoted by “ResourceKey₂” input to the analytics services manager 1702. The resource key is checked against the registered resource keys maintained by the analytics services manager 1702. Because the resource key “ResourceKey₂” matches the registered resource key represented by block 1802, control flows to block 1805 in which the stream of metric data is forwarded to corresponding metric processor 1711. The stream of metric data may also be copied to the database 1706.

The analytics services manager 1702 also manages the life cycle of each metric processor. The analytics service manager 1702 can tear down a metric processor when requested by a user and may reconstruct a metric processor when instructed by a user by resetting and replaying an historical stream of metric data stored in the database 1706.

Each metric processor updates and constructs models of metric data behavior based on a stream of metric data. The models are used to create metric data forecasts when a request for a forecast is made. As a result, each metric processor generates a real time metric data forecast in response to a forecast request. In order to generate a real time metric data forecast, each metric processor maintains the latest statistics on the corresponding stream of metric data, updates model parameters as metric data is received, and maintains a limited history of metric data. The duration of the sequence of metric data values comprising a limited history may vary, depending on the resource. For example, when the resource is a CPU or memory of single server computer, the limited history of metric comprise a sequence collected over an hour, day, or a week. On the other hand, when the resource is CPU usage or memory of an entire cluster of server computers that run a data center tenant's applications, the limited history of metric may comprise a sequence collected over days, weeks, or months. By updating the models, statistics, and maintaining only a limited history of the metric data, each metric processor utilizes a bounded memory footprint, a relatively small computational load, and computes a metric data forecast at low computational costs.

Metric data points of a metric data stream may arrive at the analytics services manager 1702 one at a time or two or more metric data points may arrive in time intervals. FIG. 19 shows an example of a limited history of metric data maintained by a metric processor 1902 of the forecast engine 1704. Plot 1904 displays data points of a limited history of metric data maintained by the metric processor 1902. For example, point 1906 represents a recently forwarded metric data value of the limited history of metric data recorded in a data-storage device 1910. The limited history of metric data is contained in a historical window 1908 of duration D. The historical window 1908 contains a sequence of metric data with time stamps in a time interval [t_(n)−D, t_(n)], where subscript n is a positive integer time index, and t_(n) is the time stamp of the most recently received metric data value z_(n) added to the limited history and in the historical window. Ideally, consecutive metric data values forwarded to the metric processor 1902 have regularly spaced time stamps with no gaps. Interpolation is used to fill in any gaps or missing metric data in the limited history of metric data. For example, square-shaped metric data point 1912 represents an interpolated metric data value in the limited history of metric data. Interpolation techniques that may be used to fill in missing metric data values include linear interpolation, polynomial interpolation, and spline interpolation. The metric processor 1902 computes statistical information and forecast model parameters based on the limited history of metric data 1904 and records the statistical information and forecast model parameters in the data-storage device 1910. The historical window 1708 advances in time to include the most recently received metric data values and discard a corresponding number of the oldest metric data values from the limited history of metric data. Plot 1914 displays data points of an updated limited history of metric data. Points 1916 and 1918 represents two recently received metric data values added to the limited history of metric data and points 1920 and 1922 that represent the oldest metric data values outside the historical window 1908 are discarded. The metric data in the historical window 1908 are called “lags” and a time stamp of a lag is called “lag time.” For example, metric data values z_(n−1) and z_(n) in the historical window are called lags and the corresponding time stamps values t_(n−1) and t_(n) and called lag times. The metric processor 1902 computes statistical information and updates model parameters stored in the data-storage device 1910 based on the latest limited history of metric data 1904.

When a forecast request is received by the metric processor 1902, the metric processor 1902 computes a metric data forecast based on the latest model parameters. The metric processor 1902 computes forecasted metric data values in a forecast interval at regularly spaced lead time stamps represented by open points. FIG. 19 shows a plot of forecasted metric data 1924 represented by open points, such as open point 1926, appended to the latest limited history of metric data. For example, a first forecasted metric data value {tilde over (Z)}_(n+1) occurs at lead time stamp t_(n+1), where “˜” denotes a forecasted metric data value.

Each metric data value in a stream of metric data may be decomposed as follows:

z _(i) =T _(i) +A _(i) +S _(i)  (1)

-   -   where         -   i=1, . . . , n;         -   n is the number of metric data values in the historical             window;         -   T_(i) is the trend component;         -   A_(i) is the stochastic component; and         -   S_(i) is the seasonal or periodic component.             Note that certain streams of metric data may have only one             component (e.g., A_(i)≠0 and T_(i)=S_(i)=0, for all i).             Other streams may have two components (e.g., A_(i)≠0,             S_(i)≠0, and T_(i)=0, for all i). And still other streams             may have all three components.

FIG. 20 shows an overview of example processing operations carried out by the metric processor 1902. The latest metric data 1914 within the historical window 1908 is input to the metric processor 1902. The historical window contains the latest sequence of metric data in the limited history. In block 2001, a trend estimate of the metric data in the historical window is computed. In decision block 2002, if the trend estimate fails to adequately fit the metric data in the historical window, the metric data is non-trendy. On the other hand, if the trend estimate adequately fits the sequence of metric data, the sequence of metric data in the historical window is trendy and control flows to block 2003 where the trend estimate is subtracted from the metric data to obtain detrended sequence of metric data over the historical window.

FIGS. 21A-21C show an example of computing a trend estimate and detrending metric data within a historical window. In FIGS. 21A-21C, horizontal axes, such as horizonal axis 2102, represent time. Vertical axes, such as vertical axis 2104, represent the amplitude range of the metric data in the historical window. In FIG. 21A, the values of the metric data represented by points, such as point 2106, vary over time, but a trend is recognizable by an overall increase in metric data values with increasing time. A linear trend may be estimated over the historical window by a linear equation given by:

T _(i) =α+βt _(i)  (2a)

where

-   -   α is vertical axis intercept of the estimated trend;     -   β is the slope of the estimated trend;     -   i=1, . . . , n; and     -   n is the time index of the most recently added metric data value         to sequence of metric data with a time stamp in the historical         window.

The index i is the time index for time stamps in the historical window. The slope α and vertical axis intercept β of Equation (2a) may be determined by minimizing a weighted least squares equation given by:

$\begin{matrix} {L = {\sum\limits_{i = 1}^{n}\; {w_{i}\left( {z_{i} - \alpha - {\beta \; t_{i}}} \right)}^{2}}} & \left( {2b} \right) \end{matrix}$

where w_(i) is a normalized weight function.

Normalized weight functions w_(i) weight recent metric data values higher than older metric data values within the historical window. Examples of normalized weight functions that give more weight to more recently received metric data values within the historical window include w_(i)=e^((i−n)) and w_(i)=i/n, for i=1, . . . , n. The slope parameter of Equation (2a) is computed as follows:

$\begin{matrix} {{\beta = \frac{\sum\limits_{i = 1}^{n}\; {{w_{i}\left( {t_{i} - t_{w}} \right)}\left( {z_{i} - z_{w}} \right)}}{\sum\limits_{i = 1}^{n}\; {w_{i}\left( {t_{i} - t_{w}} \right)}^{2}}}{where}{t_{w} = \frac{\sum\limits_{i = 1}^{n}\; {w_{i}t_{i}}}{\sum\limits_{i = 1}^{n}\; w_{i}}}{z_{w} = \frac{\sum\limits_{i = 1}^{n}\; {w_{i}z_{i}}}{\sum\limits_{i = 1}^{n}\; w_{i}}}} & \left( {2c} \right) \end{matrix}$

The vertical axis intercept parameter of Equation (2a) is computed as follows:

α=z _(w) −βt _(w)  (2d)

In other implementations, the weight function may be defined as w_(i)≡1.

A goodness-of-fit parameter is computed as a measure of how well the trend estimate given by Equation (2a) fits the metric data values in the historical window:

$\begin{matrix} {R^{2} = \frac{\sum\limits_{i = 1}^{n}\; \left( {T_{i} - z_{w}} \right)^{2}}{\sum\limits_{i = 1}^{n}\; \left( {z_{i} - z_{w}} \right)^{2}}} & (3) \end{matrix}$

The goodness-of-fit R² ranges between 0 and 1. The closer R² is to 1, the closer linear Equation (2a) is to providing an accurately estimate of a linear trend in the metric data of the historical window. In decision block 2002 of FIG. 20, when R²≤Th_(trend), where Th_(trend) is a user defined trend threshold less than 1, the estimated trend of Equation (2a) is not a good fit to the sequence of metric data values and the sequence of metric data in the historical window is regarded as non-trendy metric data. On the other hand, when R²>Th_(trend), the estimated trend of Equation (2a) is recognized as a good fit to the sequence of metric data in the historical window and the trend estimate is subtracted from the metric data values. In other words, when R²>Th_(trend), for i=1, . . . , n, the trend estimate of Equation (2a) is subtracted from the sequence of metric data in the historical window to obtain detrended metric data values:

{circumflex over (z)} _(i) =z _(i) −T _(i)  (4)

-   -   where the hat notation “{circumflex over ( )}” denotes         non-trendy or detrended metric data values.

In FIG. 21B, dashed line 2108 represents an estimated trend of the sequence of metric data. The estimated trend is subtracted from the metric data values according to Equation (4) to obtain a detrended sequence of metric data shown in FIG. 21C. Although metric data values may vary stochastically within the historical window, with the trend removed as shown in FIG. 21C, the metric data is neither generally increasing nor decreasing for the duration of the historical window.

Returning to FIG. 20, as recently forwarded metric data values are input to the metric processor 1902 and a corresponding number of oldest metric data values are discarded from the historical window, as described above with reference to FIG. 19, the metric processor 1902 updates the slope and vertical axis intercepts according to Equations (2b) and (2c), computes a goodness-of-fit parameter according to Equation (3), and, if a trend is present, subtracts the trend estimate according to Equation (4) to obtain a detrended sequence of metric data in the historical window. If no trend is present in the metric data of the historical window as determined by the goodness-of-fit in Equation (3), the sequence of metric data in the historical window is non-trendy. In either case, the sequence of metric data output from the computational operations represented by blocks 2001-2003 is called a sequence of non-trendy metric data and each non-trendy metric data value is represented by

z _(i) =A _(i) +S _(i)  (5)

where i=1, . . . , n.

The mean of the non-trendy metric data in the historical window is given by:

$\mu_{z} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\; {\hat{z}}_{i}}}$

When the metric data in the historical window has been detrended according to Equation (4) and R²>Th_(trend), the mean μ_(z)=0. On the other hand, when the metric data in the historical satisfies the condition R²≤Th_(trend), then it may be the case that the mean μ_(z)≠0.

In alternative implementations, computation of the goodness-of-fit R² is omitted and the trend is computed according to Equations (2a)-(2d) followed by subtraction of the trend from metric data in the historical window according to Equation (4). In this case, the mean of the metric data μ_(z), equals zero in the discussion below.

The sequence of detrended or non-trendy metric data may be either stationary or non-stationary metric data. Stationary non-trendy metric data varies over time in a stable manner about a fixed mean. Non-stationary non-trendy metric data, on the other hand, the mean is not fixed and varies over time. For a stationary sequence of non-trendy metric data, the stochastic process models 2004-2006 in FIG. 20 may be autoregressive moving-average models 2006-2008 (“ARMA”) computed separately for the stationary sequence of metric data in the historical window. An ARMA model is represented, in general, by

ϕ(B){circumflex over (z)} _(n)=θ(B)a _(n)  (6a)

where

-   -   B is a backward shift operator;

${\varphi (B)} = {1 - {\sum\limits_{i = 1}^{p}\; {\varphi_{i}B^{i}}}}$ ${\theta (B)} = {1 - {\sum\limits_{i = 1}^{p}\; {\theta_{i}B^{i}}}}$

-   -   a_(n) is white noise;     -   ϕ_(i) is an i-th autoregressive weight parameter;     -   θ_(i) is an i-th moving-average weight parameter;     -   p is the number of autoregressive terms called the         “autoregressive order;” and     -   q is the number of moving-average terms called the         “moving-average order;”         The white noise is a_(n) is a sequence of independent and         identically distributed random variables with mean zero and         variance σ_(a) ². The backward shift operator is defined as         B{circumflex over (z)}_(n)={circumflex over (z)}_(n−1) and         B^(i){circumflex over (z)}_(n)={circumflex over (z)}_(n−1). In         expanded notation, the ARMA model is represented by

$\begin{matrix} {{\hat{z}}_{n} = {{\sum\limits_{i = 1}^{p}\; {\varphi_{i}{\hat{z}}_{n - i}}} + a_{n} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q}\; {\theta_{i}a_{n - i}}}}} & \left( {6b} \right) \end{matrix}$

where Φ=1−ϕ₁− . . . −ϕ_(P).

The white noise parameters a_(n) may be determined at each time stamp by randomly selecting a value from a fixed normal distribution with mean zero and non-zero variance. The autoregressive weight parameters are computed from the matrix equation:

$\begin{matrix} {{\overset{\rightharpoonup}{\varphi} = {P^{- 1}\overset{\rightharpoonup}{\rho}}}{where}{{\overset{\rightharpoonup}{\varphi} = \begin{bmatrix} \varphi_{1} \\ \vdots \\ \varphi_{p} \end{bmatrix}};}{{\overset{\rightharpoonup}{\rho} = \begin{bmatrix} \rho_{1} \\ \vdots \\ \rho_{p} \end{bmatrix}};{and}}{P^{- 1} = \begin{bmatrix} 1 & \rho_{1} & \ldots & \rho_{p - 1} \\ \rho_{1} & 1 & \ldots & \rho_{p - 2} \\ \vdots & \vdots & \ddots & \vdots \\ \rho_{p - 1} & \rho_{p - 2} & \ldots & 1 \end{bmatrix}^{- 1}}} & (7) \end{matrix}$

The matrix elements are computed from the autocorrelation function given by:

$\begin{matrix} {{{\rho_{k} = \frac{\gamma_{k}}{\gamma_{0}}}{where}\gamma_{k} = {\frac{1}{n}{\sum\limits_{i = 1}^{n - k}\; {\left( {{\hat{z}}_{i} - \mu_{z}} \right)\left( {{\hat{z}}_{i + k} - \mu_{z}} \right)}}}}{\gamma_{0} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\; \left( {{\hat{z}}_{i} - \mu_{z}} \right)^{2}}}}} & (8) \end{matrix}$

The moving-average weight parameters may be computed using gradient descent. In the Example of FIG. 20, the metric processor 1902 computes three separate stochastic process models 2004-2006 for stationary sequence of non-trendy metric data in the latest historical window. For example, when the historical window of the sequence of non-trendy metric data is updated with recently received non-trendy metric data values, three sets of autoregressive and moving average weight parameters are computed for each the three ARMA models denoted by ARMA(p₁,q₁), ARMA(p₂,q₂), and ARMA(p₃,q₃).

FIG. 22 shows example weight parameters for three autoregressive moving-average models ARMA(p₁,q₁), ARMA(p₂,q₂), and ARMA(p₃,q₃). Horizontal axis 2202 represents time. Vertical axis 2204 represents a range of amplitudes of a stationary sequence of non-trendy metric data. Points, such as point 2206, represent metric data values in a historical window. FIG. 22 includes plots of three example sets of autoregressive and moving average weight parameters 2210-2212 for three different autoregressive and moving-average models. For example, ARMA model ARMA(p₃,q₃) 2212 comprises twelve autoregressive weight parameters and nine moving-average weight parameters. The values of the autoregressive weight parameters and moving-average weight parameters are computed for the stationary sequence of non-trendy metric data in the historical window. Positive and negative values of the autoregressive weight parameters and moving-average weight parameters are represented by line segments that extend above and below corresponding horizonal axes 2214 and 2216 and are aligned in time with time stamps of the non-trendy metric data.

Prior to updating the stochastic process models, when a new metric data value z_(n+1) is received by the metric processor 1902, the new metric data value is detrended according to Equation (4) to obtained detrended metric value {circumflex over (z)}_(n+1) and a corresponding estimated non-trendy metric data value {circumflex over (z)}_(n+1) ^((m)) is computed using each of the stochastic process models 2004-2006. For example, the estimated non-trendy metric data value {circumflex over (z)}_(n+1) ^((m)) may be computed using each of the ARMA models ARMA(p_(m),q_(m)) as follows:

$\begin{matrix} {{\hat{z}}_{n + 1}^{(m)} = {{\sum\limits_{i = 1}^{p_{m}}\; {\varphi_{i}{\hat{z}}_{n}}} + a_{n + 1} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{m}}\; {\theta_{i}a_{n}}}}} & (9) \end{matrix}$

where in =1, 2, 3.

Separate accumulated residual errors are computed for each stochastic model as new metric data values are received by the metric processor 1902 as follows:

$\begin{matrix} {{{Error}\left( {p_{m},q_{m}} \right)} = {\sum\limits_{i = 1}^{n}\; \left( {{\hat{z}}_{n + 1}^{(m)} - {\hat{z}}_{n + 1}} \right)^{2}}} & (10) \end{matrix}$

where

-   -   {circumflex over (z)}_(n−1) is a latest non-trendy metric data         value received by the metric processor 1902 at time stamp         t_(n+1);

{circumflex over (z)}_(n+1) ^((m)) is an estimated non-trendy metric data value computed using the ARMA model ARMA(p_(m),q_(m)) at the time stamp t_(n+1); and

-   -   ({circumflex over (z)}_(n+1) ^((m))−{circumflex over         (z)}_(n+1))² is a residual error at the time stamp t_(n+1).         After the accumulated residual error is computed, the limited         history of metric data is updated as described above with         reference to FIG. 19 and the parameters of the stochastic         process models 2004-2006 are updated.

FIG. 23 shows an example of a latest non-trendy metric data value {circumflex over (z)}_(n+1) received by the metric processor 1902 as represented by point 2306. Three candidate metric data values are separately computed using the three ARMA models ARMA(p₁,q₁), ARMA(p₂,q₂), and ARMA(p₃,q₃) as follows:

${\hat{z}}_{n + 1}^{(1)} = {{\sum\limits_{i = 1}^{p_{1}}\; {\varphi_{i}{\hat{z}}_{n}}} + a_{n + 1} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{1}}\; {\theta_{i}a_{n}}}}$ ${\hat{z}}_{n + 1}^{(2)} = {{\sum\limits_{i = 1}^{p_{2}}\; {\varphi_{i}{\hat{z}}_{n}}} + a_{n + 1} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{2}}\; {\theta_{i}a_{n}}}}$ and ${\hat{z}}_{n + 1}^{(3)} = {{\sum\limits_{i = 1}^{p_{3}}\; {\varphi_{i}{\hat{z}}_{n}}} + a_{n + 1} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{3}}\; {\theta_{i}a_{n}}}}$

where the white noise a_(n+1) is randomly selecting from the fixed normal distribution. FIG. 23 includes a magnified view 2308 of the latest non-trendy metric data value {circumflex over (z)}_(n+1) 2306 received by the metric processor 1902 and three estimated non-trendy metric data values {circumflex over (z)}_(n+1) ⁽¹⁾, {circumflex over (z)}_(n+1) ⁽²⁾ and {circumflex over (z)}_(n+1) ⁽³⁾ computed separately from the three ARMA models at the time stamp t_(n+1). Directional arrows 2311-2313 represent differences in amplitudes between the latest non-trendy metric data value {circumflex over (z)}_(n+1) 2306 and the three estimated non-trendy metric data values {circumflex over (z)}_(n+1) ⁽¹⁾, {circumflex over (z)}_(n+1) ⁽²⁾ and {circumflex over (z)}_(n+1) ⁽³⁾. Accumulated residual errors are maintained for each of the ARMA models as follows:

${{Error}\left( {p_{1},q_{1}} \right)} = {\sum\limits_{i = 1}^{n}\; \left( {{\hat{z}}_{n + 1}^{(1)} - {\hat{z}}_{n + 1}} \right)^{2}}$ ${{Error}\left( {p_{2},q_{2}} \right)} = {\sum\limits_{i = 1}^{n}\; \left( {{\hat{z}}_{n + 1}^{(2)} - {\hat{z}}_{n + 1}} \right)^{2}}$ and ${{Error}\left( {p_{3},q_{3}} \right)} = {\sum\limits_{i = 1}^{n}\; \left( {{\hat{z}}_{n + 1}^{(3)} - {\hat{z}}_{n + 1}} \right)^{2}}$

Returning to FIG. 20, when a forecast is requested 2007 in block 2008, the accumulated residual errors of the stochastic models are compared and the stochastic process model with the smallest accumulated residual error is selected for forecasting. For example, the ARMA model ARMA(p_(m),g_(m)) may be used to compute forecasted metric data values as follows:

$\begin{matrix} {{\hat{z}}_{n + l}^{(m)} = {{\sum\limits_{i = 1}^{l - 1}\; {\varphi_{i}{\hat{z}}_{n + i - 1}^{(m)}}} + {\sum\limits_{i = l}^{p_{m}}\; {\varphi_{i}{\hat{z}}_{n + l - i}}} + a_{n + l} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{m}}\; {\theta_{i}a_{n + l - i}}}}} & (11) \end{matrix}$

where

-   -   l=1, . . . , L is a lead time index with L the number of lead         time stamps in the forecast interval;     -   {circumflex over (z)}_(n) ^((m)) is zero; and     -   a_(n+1) is the white noise for the lead time stamp t_(n+1).

FIG. 24 shows forecasted metric data values computed using weight parameters of the ARMA model 2012 ARMA(p₃,q₃) in FIG. 22. In the example of FIG. 24, horizontal axis 2402 is a time axis for positive integer lead time indices denoted by 1. The first three forecasted metric data values, denoted by “x's” in FIG. 24, are computed using ARMA(p₃,q₃) as follows:

${\hat{z}}_{n + 1}^{(3)} = {{\sum\limits_{i = 1}^{p_{3}}\; {\varphi_{i}{\hat{z}}_{n + 1 - i}}} + a_{n + 1} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{3}}\; {\theta_{i}a_{n + l - i}}}}$ ${\hat{z}}_{n + 2}^{(3)} = {{\varphi_{1}{\hat{z}}_{n + 1}^{(3)}} + {\sum\limits_{i = 1}^{p_{3}}\; {\varphi_{i}{\hat{z}}_{n + 2 - i}}} + a_{n + 2} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{3}}\; {\theta_{i}a_{n + l - i}{and}}}}$ ${\hat{z}}_{n + 3}^{(3)} = {{\varphi_{1}{\hat{z}}_{n + 2}^{(3)}} + {\varphi_{2}{\hat{z}}_{n + 1}^{(3)}} + {\sum\limits_{i = 3}^{p_{3}}\; {\varphi_{i}{\hat{z}}_{n + 3 - i}}} + a_{n + 3} + {\mu_{z}\Phi} + {\sum\limits_{i = 1}^{q_{3}}\; {\theta_{i}a_{n + l - i}}}}$

In still other implementations, the stochastic process models 2004-2006 in FIG. 20 may be implemented as autoregressive process (“AR”) models given by:

$\begin{matrix} {{\hat{z}}_{n} = {{\sum\limits_{i = 1}^{p}\; {\varphi_{i}{\hat{z}}_{n - i}}} + a_{n} + {\mu_{z}\Phi}}} & (12) \end{matrix}$

The autoregressive process model is obtained by omitting the moving-average weight parameters form the ARMA model. By omitting the moving-average model, computation of the autoregressive weight parameters of the autoregressive model is less computationally expensive than computing the autoregressive and moving-average weight parameters of the ARMA models. When the historical window of the sequence of non-trendy metric data is updated with recently received non-trendy metric data values, three sets of autoregressive weight parameters are computed for each the three AR models denoted by AR(p₁), AR(p₂), and AR(p₃). Accumulated residual errors are maintained for each of the AR models. Forecasted metric data values {circumflex over (z)}_(n+1) ^((m)) are computed for lead times using Equation (11) with the moving-average weight parameters equal to zero and the AR model with smallest accumulated residual error at the time of the forecast request.

Unlike a stationary sequence of non-trendy metric data, a non-stationary sequence of non-trendy metric data does not vary over time in a stable manner about a fixed mean. In other words, a non-stationary sequence of non-trendy metric data behaves as the though the metric data values of the sequence have no fixed mean. In these situations, one or more of the stochastic process models 2004-2006 in FIG. 20 may be implemented using an autoregressive integrated moving-average (“ARIMA”) model given by:

ϕ(B)∇^(d) {circumflex over (z)} _(n)=θ(B)a _(n)  (13)

where ∇^(d)=(1−B)^(d). The ARIMA autoregressive weight parameters and move-average weight parameters are computed in the same manner as the parameters of the ARMA models described above. The ARIMA model, denoted by ARIMA(p₁,q₁), ARIMA(p₂,q₂), and ARIMA(p₃,q₃), with the smallest accumulated residual error at the time of the forecast request is used to compute forecasted metric data values {circumflex over (z)}_(n+1) ^((m)) for lead times in the forecast interval.

Returning to FIG. 20, certain streams of metric data may have pulse wave patterns. Other streams of metric data may have a single time varying periodic pattern or a combination of period patterns, such as hourly, daily, weekly or monthly periodic patterns, and are called “seasonal.” Other streams of metric data may not be periodic. Because pulse wave metric data is a special type of periodic data, in decision block 2009, edge detection is used to determine if the sequence of non-trendy metric data in the historical window is pulse wave metric data. If edge detection reveals that the metric data is pulse wave metric data, control flows to determining the pulse wave model 2010. Otherwise, control flows to block 2011 to determine if the metric data contains a seasonal pattern. Seasonality in a sequence of non-trendy metric data is a regular periodic pattern of amplitude changes that repeats in time periods. A seasonal period is determined in a seasonal model in block 2011.

FIG. 25 shows a plot of an example stream of metric data 2500. Horizontal axis 2502 represents time. Vertical axis 2504 represents a range of amplitudes for metric data values. The stream of metric data comprises pulses 2506-2510 separated by low amplitude time intervals 2511-2514. The stream of metric data may represent network traffic, memory usage, or CPU usage for a server computer that runs a periodically executed VM. The low amplitude time intervals 2511-2514 represent time intervals in which the VM is idle. Pulses 2506-2510 represent time intervals when the VM is running. This stream of metric data is an example of metric data modeled using a pulse wave model 2010.

FIG. 26 shows a plot of an example stream of metric data 2600 that exhibits two seasonal periods. Horizontal axis 2602 represents time. Vertical axis 2604 represents a range of amplitudes for metric data values. Oscillating curve 2606 represents a stream of metric data with two seasonal periods. A first longer seasonal period appears with regularly spaced larger amplitude oscillations 2606-2609 separated by regularly spaced smaller amplitude oscillations 2610-2613. A second shorter seasonal period exhibits oscillations over much shorter time intervals. This stream of metric data is an example of seasonal metric data modeled using the seasonal model 2011.

In block 2009 of FIG. 20, edge detection is applied to the metric data in the historical window. An exponentially weighted moving average (“EWMA”) of absolute differences between two consecutive non-trendy metric data values denoted by |{circumflex over (z)}_(i)−{circumflex over (z)}_(i−1) is maintained for i=1, . . . , n metric data values in the historical window. The EWMA for the latest time stamp t_(n) in the historical window is computed recursively by:

MA _(n) =α|{circumflex over (z)} _(n) −z _(n−1)|_(n)+(1−αMA _(n−1)  (14a)

where

-   -   MA₀=0; and     -   0<α<1.         For example, the parameter a may be set 0.1, 0.2, or 0.3. For         each new non-trendy metric data value {circumflex over         (z)}_(n+1), the absolute difference |{circumflex over         (z)}_(n+1)−{circumflex over (z)}_(n)| is computed. The new         non-trendy metric data value {circumflex over (z)}_(n+1) is a         spike in the magnitude of the stream of metric data, when the         absolute difference satisfies the following spike threshold         condition:

|{circumflex over (z)} _(n+1) −{circumflex over (z)} _(n) |>Th _(spike)  (14b)

where Th_(spike)=C×MA_(n).

The parameter C is a numerical constant (e.g., C=4.0, 4.5, or 5.0). When the absolute difference Δ_(n+10) is less than the spike threshold, control flows to seasonal model in block 2011 of FIG. 20. When the new non-trendy metric data value {circumflex over (z)}_(n+1) satisfies the condition given by Equation (14b), edge detection is applied to determine if sequence of non-trendy metric data comprises pulse wave edges in a backward time window [{circumflex over (z)}_(n−x), {circumflex over (z)}_(n)] and a forward time window [{circumflex over (z)}_(n), {circumflex over (z)}_(n+X)], where X is a fixed number of metric data points.

FIGS. 27A-27B show edge detection applied to a sequence of metric data. FIG. 27A shows a plot of a square pulse 2702 in metric data. The metric data may have been smoothed to suppress the noise using a smoothing filter applied to the metric data in the historical window. The smoothing filter may be a moving median filter, a moving average filter, or a Gaussian filter. Horizontal axis 2704 represents a time window of the historical window. For example, the time window may be 20 minutes, 30 minutes, or 40 minutes or a longer time window of the historical window. Vertical axis 2706 represents a range of metric data amplitudes. Metric data are averaged in two adjacent sliding time windows within the larger time window 2704. The two adjacent time windows are referred to as a previous time window, denoted by w_(p), and a next time window, denoted by w_(n). FIG. 27A shows examples of adjacent sliding previous and next time windows. For example, directional arrows 2708 and 2710 represent adjacent previous and next time windows, respectively. In the example of FIG. 27A, the previous and next time windows each contain five metric data points. In other implementations, the previous and next time windows may contain any number of metric data points. FIG. 27A shows the locations of previous and next time windows as the windows move within the larger time window 2704 represented by horizontal time axis 2704. Each next time window becomes a previous time window. For example, next time window 2710 becomes previous time window 2712. For each location of the previous time window, an average of the metric data values in the previous time window is computed as follows:

$\begin{matrix} {{\Sigma \; {S\left( w_{p} \right)}} = {\frac{1}{{S\left( w_{p} \right)}}{\sum\limits_{z_{i} \in {S{(w_{p})}}}z_{i}}}} & \left( {15a} \right) \end{matrix}$

where

-   -   S(w_(p)) is the set of metric data with time stamps in the         previous time window w_(p); and     -   |S(w_(p))| is the number of metric data points with time stamps         in the previous time window w_(p).         An average of the metric data values in the next time window is         computed as follows:

$\begin{matrix} {{\Sigma \; {S\left( w_{n} \right)}} = {\frac{1}{{S\left( w_{n} \right)}}{\sum\limits_{z_{i} \in {S{(w_{n})}}}z_{i}}}} & \left( {15b} \right) \end{matrix}$

where

-   -   S(w_(n)) is the set of metric data with time stamps in the next         time window w_(n); and     -   |S(w_(n))| is the number of metric data values with time stamps         in the next time window w_(n).

The average metric data ΣS(w_(p)) in the previous time window and the average metric data ΣS(w_(n)) in the next time window are moving averages as the previous and next time windows are moved within the larger time window 2704. A change parameter is computed for each location of the previous and next time windows in the larger time window 2704 as follows:

Δ=|ΣS(w _(p))−ΣS(w _(n))|  (16)

FIG. 27A shows a plot of change parameters for each location of the previous and next time windows. Horizontal axis 2716 represents time. Vertical axis 2718 represents a range of change parameter values. For example, solid point 2720 represents the value of a change parameter for the metric values in previous and next time windows 2708 and 2710. Solid point 2722 represents the value of a change parameter for the metric values in previous and next time windows 2712 and 2714.

When the following two conditions are satisfied for a change parameter, the corresponding previous and next time windows are a time interval, denoted by w_(p)∪w_(n), that contains an edge of a square pulse of metric data. The first condition is given by

Δ>Th ₁×(max{z _(i)}−min{z _(i)})  (17a)

where

-   -   0<Th₁<1 (e.g., Th₁=0.05);     -   max{z_(i)} is the maximum metric value in set of metric data         values in the historical window;     -   min{z_(i)} is the minimum metric value in the set of metric data         values in the historical window; and     -   max{z_(i)}−min{z_(i)} is the maximum amplitude range of metric         data values in the historical window.

The quantity Th₁×(max{z_(i)}−min{z_(i)}) is the maximum amplitude range of metric data values threshold. The second condition is given by

Δ>Th ₂ ×MA(Δ)  (17b)

where

-   -   0<Th₂<1 (e.g., Th₂=0.25); and     -   MA(Δ) is the EWMA.         The quantity Th₂×MA(Δ) is a change parameter EWMA threshold. The         EWMA of the change parameters is a weighted average that assigns         weight to the most recently generated change parameter relative         to change parameters generated in the past and is given by:

MA(Δ_(i))=λΔ_(i)+(1−λ)MA(Δ_(i−1))

where

-   -   MA(Δ₀) is the mean of historical change parameters;     -   Δ_(i) is the change parameter of the i-th most recent previous         and next time window; and     -   0<λ<1 is a constant that determines the weight at which older         change parameters enter into calculation of the EWMA (i.e., a         large value of λ gives more weight to recent change parameters         and less weight to older change parameters).

In FIG. 27A, dashed lines 2724 and 2726 represent thresholds in Equations (17a) and (17b). Change parameters 2728 and 2730 satisfy the two conditions. Change parameter 2730 satisfies one condition but not the other. Change parameters with values less than dashed line 2724 do not satisfy either condition. Because the change parameters 2728 and 2730 satisfy the two conditions, the corresponding previous and next time windows 2734 and 2736 are time intervals that contain edges of the square pulse 2702.

An absolute difference is computed between each pair of consecutive metric data values with time stamps in a time interval w_(p)∪w_(n) identified as having an edge. The largest absolute difference between consecutive pairs of metric data values in the time interval is given by

|z _(i+1) −z _(i)|=max{|z _(j+1) −z _(j)|}_(j=1) ^(j−1)  (18)

where

-   -   z_(j)∈w_(p)∪w_(n) for j=1, . . . ,J; and     -   J is the number of metric data points in the time interval         w_(p)∪w_(n).         When z_(i+1)−z_(i)>0, the edge is an upward or increasing edge         and time t_(i+1) is the time of the upward edge. When         z_(i+1)−z_(i)<0, the edge is a downward or decreasing edge and         time t_(i) is the time of the downward edge.

FIG. 27B shows a plot of the square pulse 2702 of metric data values and time intervals 2738 and 2740 that correspond to previous and next time intervals 2734 and 2736, respectively. Line segments between metric data points represent absolute differences. Line segment 2742 between metric data values 2744 and 2746 is the largest difference between consecutive pairs of metric data points in the time interval 2738. Line segment 2748 between metric points 2750 and 2752 is the largest difference between consecutive pairs of metric data values in the time interval 2740. Time stamps 2754 and 2756 of the metric data values 2746 and 2752 are the edges of the square pulse 2702.

Returning to FIG. 20, the pulse wave model 2010 estimates the pulse width and period for the pulse wave stream of metric data. The pulse width can be estimated as a difference in time between consecutive upward and downward edges. The period can be estimated as a difference in time between two consecutive upward (or downward) edges.

FIG. 28 shows pulse widths and periods of the stream of metric data 2500. Each edge has a corresponding 3-tuple (t_(s), A, sign), where t_(s) represents the time stamp of the pulse edge, A is the amplitude of the pulse edge, and “sign” indicates whether pulse is upward denoted by “+” or downward denoted by Pulse widths denoted by pw₁, pw₂, pw₃, pw₄, and pw₅ are computed as a difference between time stamps of consecutive upward and downward edges. Periods denoted by p₁, p₂, p₃, p₄, and p₅ are computed as a difference between time stamps of two consecutive upward (or downward) edges. The latest pulse widths and periods are recorded in corresponding circular buffer back-sliding histograms described below with reference to FIG. 30A.

Returning to FIG. 20, if the sequence of non-trendy metric data is not pulse-wave metric data, the metric data may be seasonal metric data and a seasonal period is determined in seasonal model 2011. The seasonal model 2011, begins by applying a short-time discrete Fourier transform (“DFT”) given by:

$\begin{matrix} {{Z\left( {m,{k/N}} \right)} = {\sum\limits_{i = 1}^{N}\; {{\hat{z}}_{i}\mspace{14mu} {w\left( {i - m} \right)}{\exp \left( {{- j}\; 2\pi \; {{ki}/N}} \right)}}}} & (19) \end{matrix}$

where

-   -   m is an integer time shift of a short-time window;     -   j is the imaginary constant √{square root over (−1)};     -   k=0, . . . , N−1 is a frequency spectrum sample;     -   N is the number of data points in a subset of the historical         window N≤n); and     -   w(i−m) is a window function.         The window function w(i−m) is function that tapers toward both         ends of the short-time window. For example, the window function         can be a Hann function, a Hamming function, or Gaussian         function. The spectrum Z(m, k/N) is a complex valued function of         m and k. The power spectral density (“PSD”) is given by:

$\begin{matrix} {{{{PSD}\left( {m,{k/N}} \right)} = {{Z\left( {m,{k/N}} \right)}}^{2}}{where}{{k = 0},\ldots \;,{{N/2};}}{{f_{k} = {\frac{2k}{N}f_{c}}};{and}}{f_{c}\mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {Nyquist}\mspace{14mu} {{frequency}.}}} & (20) \end{matrix}$

The PSD is the power of the frequency spectrum at N/2+1 frequencies. The PSD values PSD (m, k/N) fault a periodogram over a domain of frequency samples k (i.e., f_(k)) for each time shift in.

The short-time DFT may be executed with a fast Fourier transform (“FFT”). Ideally, a high-resolution FFT comprising a large window size and high sampling rate would be used to compute a PSD in each historical window of the FFT to provide complete frequency spectrum information in the historical window. By finding a maximum PSD point at each time shift M and curve fitting, various seasonal patterns and reconstructed metric data values can ideally be forecasted with an inverse FFT. However, computing a high-resolution FFT and storing the full PSD for a sequence of non-trendy metric data is computationally expensive and time consuming in a resource constrained management system that already receives thousands of different streams of metric data and in which real time forecasts are needed to respond to rapidly to changing demands for computational resources in a distributed computing system.

Methods described herein avoid the slowdown created by a high-resolution FFT by:

-   -   1) using an FFT in a short-time window with a small number of         metric data points (e.g., a short-time window may have N=64, 128         or 256 sequential non-trendy metric data points of the limited         history) for three different coarse sampling rates,     -   2) extracting a single principle frequency from each PSD and         tracking a most recent mode of the principle frequency, and     -   3) performing a local auto-correlation function (“ACF”) search         in the time domain to refine estimation of a principle period         that corresponds to the principle frequency of the metric data         to compensate for resolution lost with coarse sampling rates and         spectral leakage.

The FFT is applied to subsequences of the sequence of non-trendy metric data, each subsequence comprising N metric data points sampled from the sequence of non-trendy metric data using a different sampling rate. Each subsequence of metric data points is searched for a periodic pattern. For example, the example sequence of non-trendy metric data 2600 shown in FIG. 26 appears to have a short periodic pattern and a longer periodic pattern as described above with reference to FIG. 26. The period determined for the shorter sampling rate has higher priority in forecasting than a period obtained for a longer sampling rate.

FIG. 29 shows a bar graph 2900 of four different examples of sampling rates and associated with different subsequences of samples of the same sequence of non-trendy metric data. Horizontal axis 2902 represent time in hours. Hash-marked bars 2904-2907 represent durations of three different sampling rates applied to the same stream of metric data to collect three different subsequences of non-trendy metric data over three different time intervals. Each subsequence contains N=64 sequential non-trendy metric data points of the larger sequences of non-trendy metric data. Plots 2908-2911 are example plots of subsequences of metric data resampled from the sequence of non-trendy metric data over four different time intervals and the four different sampling rates represented by the bar graph 2700. In plots 2908-2911, horizontal axes 2912-2915 represent different time intervals. Time zero along each axis represents the current time. In plot 2908, horizontal axis 2912 represents a time interval of 16 hours. Curve 2916 represents a subsequence of metric data sampled from the sequence of non-trendy metric data over a 16-hour time interval at the sampling rate of 15 minutes. In plot 2908, horizontal axis 2913 represents a time interval of 64 hours. Curve 2918 represents a subsequence of metric data sampled from the sequence of non-trendy metric data over a 64-hour time interval at the sampling rate of 1 hour. In plot 2910, horizontal axis 2914 represents a time interval of 16 days. Curve 2919 represents a sequence of metric data sampled from the sequence of non-trendy metric data over a 16-day time interval at the sampling rate of 6 hours. In plot 2911, horizontal axis 2915 represents a time interval of 64 days. Curve 2920 represents metric data sampled from the sequence of non-trendy metric data over a 64-day time interval at the sampling rate of 24 hours. The different sampling rates applied to the same sequence of non-trendy metric data over different time intervals reveal different frequency patterns or seasonal periods within the sequence of non-trendy metric data. Subsequences of metric data 2918 exhibits a seasonal period. Subsequences of metric data 2916, 2919, and 2920 exhibit no discernible periodic pattern. If it is the case that different periods are present in two or more subsequences of metric data, the period associated with the shorter sampling rate has higher priority in forecasting than the period associated with the longer sampling rate. The dominant frequency or period of the subsequence of metric data with shortest associated sampling rate is determined. Methods for determining the dominant frequency or period of a sequence of non-trendy metric data are described below with reference to FIGS. 30-32.

FIG. 30 shows an example of periodograms computed for a series of short-time windows of a sequence of non-trendy metric data. In FIG. 30, horizontal axis 3002 represents time. Vertical axis 2804 represents a range of metric data amplitudes. Curve 3006 represents a sequence of non-trendy metric data. Brackets 3008-3011 represents the location of a moving overlapping short-time window of non-trendy metric data as non-trendy metric data is received by the seasonal model 2011. For each short-time window, an FFT is applied to a small number N of the latest metric data points followed by computation of a PSD. For example, short-time window 3008 contains a subsequence of non-trendy metric data values up to a current time t₀. An FFT 3012 is applied to the latest subsequence of metric data (e.g., N=64) in the shot-time window 3008 followed by computation of a PSD 3014. As more metric data is received and sampled at the sampling rate, the FFT is applied to the subsequence of metric data in a current short-time window followed by computation of a PSD. For example, short-time window 3009 contains a subsequence of metric data up to a next current time t₁. An FFT 3016 is applied to the subsequence of latest metric data (e.g., N=64) in the shot-time window 3009 followed by computation of a PSD 3018. FIG. 30 also shows example plots of periodograms 3020-3023 for each the PSDs computed from the subsequences of metric data in each of the corresponding short-time windows 3008-3011. Axis 3026 represents a range of frequencies. Axis 3028 represents a range of time shifts. Axis 3030 represents a range of power.

For each periodogram, an approximate area of the periodogram is computed. For example, the approximate area of a periodogram can be computed using the Trapezoid Rule:

$\begin{matrix} {{PSD}_{Area} = {\frac{N}{2}{\sum\limits_{k = 1}^{N/2}\; \left( {{{PSD}\left( \frac{k - 1}{N} \right)} - {{PSD}\left( \frac{k}{N} \right)}} \right)}}} & (21) \end{matrix}$

Other methods may be used to compute the area of a periodogram, such as Simpson's rule and Romberg's method. Candidate principle frequencies of a periodogram are identified from the approximate area of the periodogram using the following threshold condition:

$\begin{matrix} {{{\underset{k}{argmax}\left\{ \left( {{{PSD}\left( \frac{k}{N} \right)}*{K_{trap}\left( \frac{k}{N} \right)}} \right) \right\}} > {{Th}_{princ}*{\sum\limits_{k = 0}^{N/2}{{PSD}\left( \frac{k}{N} \right)}}}}{where}{{\,_{\;}^{``}{*_{\;}^{"}\mspace{14mu} {means}\mspace{14mu} {convolution}}};}{{{Th}_{princ} = {{PSD}_{Area}\text{/}Q}};{and}}{{K_{trap}\left( \frac{k}{N} \right)} = {{C_{1}{{PSD}\left( \frac{k - 1}{N} \right)}} + {C_{2}{{PSD}\left( \frac{k}{N} \right)}} + {C_{3}{{PSD}\left( \frac{k + 1}{N} \right)}}}}} & (22) \end{matrix}$

The parameter Q is a positive integer (e.g., Q=3, 4, or 5) and K_(trap)(k/N) is called a normalized three-point trapezoid window. The parameters C₁, C₂, and C₃ are normalized to 1. For example, C₁=C₃=0.25 and C₂=0.5. If none of the frequencies of the periodogram satisfies the condition given by Equation (19), the subsequence of the sequence of non-trendy metric data does not have a principle frequency in the short-time window of the FFT and is identified as non-periodic.

FIG. 31A show a plot of the periodogram 3021 shown in FIG. 30.

Horizontal axis 3102 represents a frequency spectrum sample domain. Vertical axis 3104 represents a power range. Curve 3106 represents the power spectrum present in the subsequence of metric data over a spectral domain of frequencies k/N. The area under the curve 3106 may be approximated by Equation (21). Dashed line 3108 represents the principle frequency threshold Th_(princ). In this example, the periodogram reveals two strong peaks 2910 and 3112 above the threshold 3108 with corresponding frequencies k₁/N and k₂/N. However, which of the two peaks 3110 and 3112 is the principle frequency cannot be determined directly from the periodogram alone.

Each PSD value PSD (k/N) of a periodogram is the power in the spectral domain at a frequency k/N or equivalently at a period N/k in the time domain. Each DFT bin corresponds to a range of frequencies or periods. In particular, Z(k/N) bin corresponds to periods in the time interval

$\left\lbrack {\frac{N}{k},\frac{N}{k - 1}} \right).$

The accuracy of discovered candidate principle frequencies based on the periodogram deteriorates for large periods because of the increasing width of the DFT bins (N/k). In addition, spectral leakage causes frequencies that are not integer multiples of the DFT bin width to spread over the entire frequency spectrum. As a result, a periodogram may contain false candidate principle frequencies. However, a periodogram may provide a useful indicator of candidate principle frequencies.

In certain implementations, the principle frequency of the periodogram is determined by computing an autocorrelation function (“ACF”) within each neighborhood of candidate periods that correspond to candidate principle frequencies of the periodogram. The autocorrelation function over time lags T is given by:

$\begin{matrix} {{{ACF}(\tau)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{{\hat{z}}_{i}{\hat{z}}_{i + \tau}}}}} & (23) \end{matrix}$

The ACF is time-domain convolution of the subsequence of non-trendy metric data values {circumflex over (z)}_(i) in the short-time window of the FFT. Given the candidate principle frequencies of the periodogram that satisfy the threshold requirements of the condition in Equation (22), the ACF is used to determine which of the corresponding candidate periods in the time domain is a valid principle period. A candidate period with an ACF value located near a local maximum of the ACF (i.e., located within a concave-down region) is a valid period. A candidate period with an ACF value located near a local minimum of the ACF (i.e., located within a concave-up region) is not a valid period and is discarded. For a period with an ACF value that lies on a concave-down region of the ACF, the period is refined by determining the period of a local maximum ACF value of the concave-down region. The period of the local maximum is the principle period used to forecast seasonal metric data.

FIG. 31B shows a plot of an example ACF that corresponds to the periodogram shown in FIG. 31A. Horizontal axis 3114 represents time. Vertical axis 3016 represents a range of ACF values. Dashed curve 3118 represents ACF values computed according to Equation (23) over a time interval. Periods N/k₁ and N/k₂ represent candidate periods that correspond to candidate principle frequencies k₂/N and k₁/N in FIG. 31A. Open points 3120 and 3122 are ACF values at candidate periods N/k₁ and N/k₂. Rather than computing the full ACF represented by dashed curve 3118 over a large time interval, in practice, the ACF may be computed in smaller neighborhoods 3124 and 3126 of the candidate periods as represented by solid curves 3128 and 3130. The ACF value 3122 is located on a concave-down region of the ACF and corresponds to the largest of the two candidate principle frequencies. The other ACF value 3120 is located on a concave-up region of the ACF and corresponds to the smallest of the two candidate principle frequencies.

A neighborhood centered at the candidate period N/k is represented by:

$\begin{matrix} {{NBH}_{N/k} = \left\lbrack {a,\ldots \;,\frac{N}{k},\ldots \;,b} \right\rbrack} & (24) \end{matrix}$

In certain implementations, the end points for the neighborhoods may be given by:

$a = {{\frac{1}{2}\left( {\frac{N}{k + 1} + \frac{N}{k}} \right)} - 1}$ and $b = {{\frac{1}{2}\left( {\frac{N}{k} + \frac{N}{k - 1}} \right)} + 1}$

The upward or downward curvature of the ACF in the neighborhood of a candidate period is determined by computing a linear regression model for a sequence of points t between the endpoints of the neighborhood NBH_(N/k). A split period within the search interval R_(N/k) is obtained by minimizing a local approximation error for two line segments obtained from linear regression as follows:

$\begin{matrix} {t_{split} = {\arg \; {\min\limits_{P}\left( {{{error}\left( {S\left( {a,t} \right)} \right)} + {{error}\left( {S\left( {{t + 1},b} \right)} \right)}} \right)}}} & (25) \end{matrix}$

where

-   -   t is point in the neighborhood NBH_(N/k);     -   S(a, t) is a first line segment fit to points between point a         and point t of the search interval NBH_(N/k);     -   S(t+1, b) is a second line segment fit to points between point         t+1 and point b of the search interval NBH_(N/k);     -   error (S(a, t)) is the error between the S(a, t) and ACF values         between point a and point t; and     -   error (S(t+1, b)) is the error between S(t+1, b) and ACF values         between point t+1 and point b.         If the slopes of the first line segment S(a, t_(split)) and the         second line segment S(t_(split)+1, b) are correspondingly         negative and positive, then the ACF value is in a concave-up         region of the ACF and the corresponding period is discarded. If         the slopes of the first line segment S(a, t_(split)) and second         line segment S(t_(split)+1, b) are correspondingly positive and         negative, then the ACF value is in a concave-down region of the         ACF and the corresponding candidate period is kept. Once a         candidate period of a concave-down region has been identified,         the local maximum ACF may be located at the end point of the         first line segment S(a, t_(split)) or located at the start point         of the second line segment S(t_(split)+1, b). Alternatively, a         hill-climbing technique, such as gradient ascent, is applied to         determine the local maximum ACF of the concave-down region. The         period that corresponds to the ACF local maximum is the         principle period and is seasonal parameter used to forecast         seasonal metric data over a forecast interval.

FIG. 31C shows examples of line segments computed from ACF values in the neighborhoods 3124 and 3126. First and second line segments 3132 and 3134 in the neighborhood 3124 have negative and positive slopes, respectively. As a result, the candidate period N/k₂ is in a concave-up region of the ACF and is discarded. On the other hand, first and second line segments 3136 and 3138 in the neighborhood 3126 have positive and negative slopes, respectively. As a result, the candidate period N/k₁ is in a concave-down region of the ACF. The local maximum 3140 with principle period N/k* may be at the end of the first line segment or beginning of the second line segment or determined by applying a hill-climbing technique. The principle period is a seasonal parameter.

In other implementations, rather than checking each candidate period of the candidate frequencies that satisfy the condition in Equation (24) in neighborhoods of the ACF, only the candidate period that corresponds to the largest candidate principle frequency is checked using the ACF to determine if the candidate period is a principle period.

Recent mode tracking may be used to determine robust periodic model parameter estimates. Recent mode tracking is implemented with a circular buffer back-sliding histogram to track recent distributions. The periodic parameters are stored in a circular buffer. When a latest periodic parameter is determined, the periodic parameter is input to the circular buffer to overwrite the oldest periodic parameter stored in the buffer. The back-sliding histogram is updated by incrementing the count of the histogram bin the latest periodic parameter belongs to and by decrementing the count of the histogram bin the oldest periodic parameter belongs to. The mode tracker outputs the histogram bin with the largest count when the count is greater than a histogram threshold defined as Th_(hist)=C×total_count, where 0<C<1 (e.g., C equals 0.2, 0.3, 0.4, or 0.5) and total_count is the total count of periodic parameters recorded in the histogram. For each histogram bin, the count of periodic parameters in the histogram bin, denoted by Count(bin), is compared with the histogram threshold. When the following condition is satisfied

Count(bin)>Th _(hist)  (26)

the latest periodic parameter with a count added to the bin with Count(bin) that satisfies Equation (26) is used to forecast periodic metric data. On the other hand, if none of the counts of the histogram bins are greater than the histogram threshold, then forecasting of the metric data is not carried out with any of the periodic parameters of the histogram bins and the metric data in the historical window does not have a periodic pattern.

FIG. 32A shows a plot of an example histogram of periodic parameters for the pulse wave model. Horizontal axis 3202 represents a time bin axis. Vertical axis 3204 represents counts. In FIG. 32A, histogram 3206 represents a back-sliding histogram of pulse widths and histogram 3208 represents a back-sliding histogram of periods for pulse-wave metric data for seasonal model. Dashed line 3210 represents a histogram threshold for pulse widths. Dashed line 3212 represents a histogram of threshold for periods. In the example of FIG. 32A, the count of pulse widths in histogram bin 3214 is greater than the histogram threshold 3210 and the count of periods in histogram bin 3216 is greater than the histogram threshold 3212. In this case, the most recent pulse width and period counted in corresponding historical bins 3214 and 3216 are pulse wave period parameters used to forecast pulse wave metric data.

The principle frequency computed for each short-time interval as described above with reference to FIGS. 31-31 is recorded in a principle frequency back-sliding circular buffer that maintains a fixed number of the most recently generated principle frequencies. A count for each principle frequency, denoted by Count(n), where f_(i) is the i-th principle frequency (i.e., f_(i)=k_(i)/N), is maintained. When a new principle frequency is computed for a most recent short-time interval and added to the principle frequency circular buffer, the oldest principle frequency stored in the buffer is deleted. The count of the new principle frequency is incremented and the count of the oldest principle frequency removed from the buffer is decremented. The counts of the principle frequencies are rank ordered from largest to smallest. The principle frequency with the largest count that satisfies the following condition is the dominant principle frequency and the corresponding principle period is used to forecast seasonal data:

Count₁(f _(i))>N _(f)×Count₂(f _(j))  (27)

where

-   -   Count₁(f_(i)) is the largest count;     -   Count₂(f_(i)) is the second largest count; and     -   N_(f) is threshold count factor (e.g., N_(f) equals 2, 3, or 4).         The quantity N_(f)×Count₂(f_(j)) is called a dominant principle         frequency threshold. If the condition given by Equation (27) is         not satisfied, then there is no seasonal pattern in the metric         data.

FIG. 32B shows principle frequencies and an example histogram 3218 of principle frequencies. Blocks 3221-3224 represent principles frequencies computed for each of the short-time windows 3008-3011 as described above with reference to FIGS. 30-31. Horizontal axis 3226 represents frequencies. Vertical axis 3228 represent counts. Each dot represents one count for a principle frequency. For example, dot 3230 represents principle frequency f₄ generated for the short-time window 3010. Dot 3232 represents the most recently computed principle frequency f₂ generated for the short-time window 3011. Clearly principle frequency f₄ occurs with the largest count of Count₁(f₄)=28 and principle frequency f₅ occurs with the second largest count of Count₁(f₅)=8. For a threshold principle frequency factor N_(f)=3, the count Count₁(f₄) satisfies the condition given by Equation (27) and the principle frequency f₄ (i.e., principle period p₄=1/f₄) is considered the dominant frequency and is used for forecasting as described below.

In another implementation, the i-th principle frequency f_(i) is identified as the dominant principle frequency and is used for seasonal forecasting when the following condition is satisfied

$\begin{matrix} {{{PSD}\left( f_{i} \right)} > {N_{p} \times {\sum\limits_{j}{{PSD}\left( f_{j} \right)}}}} & (28) \end{matrix}$

where

-   -   PSD (f_(i)) is the power spectral density value for the         principle frequency f_(i);     -   PSD (f_(j)) is power spectral density value for the principle         frequency f_(j); and     -   N_(p) is threshold spectral density factor (e.g., N_(p) equals a         ½, ⅓, or ¼).         In other words, when the power spectral density of the principle         frequency f is greater that fraction of the total power spectral         density of the principle frequencies, then the principle         frequency f_(i) is the dominant principle frequency and is used         for forecasting as described below. If the condition given by         Equation (28) is not satisfied for any of the principle         frequencies stored in the principle frequency circular buffer,         then there is no seasonal pattern in the metric data.

Returning to FIG. 20, junction 2012 represents determining appropriate models for forecasting metric data over a forecast interval executed in block 2013. Let {tilde over (z)}_(n+1) represent forecasted metric data values for lead times t_(n+1) in a forecast interval with l=1, . . . , L. The following three conditions are considered in determining at appropriate model in junction 1812 for computing a forecast over a forecast interval in block 2013:

(1) Metric data in the historical window may not have a pulse wave pattern or a seasonal period. In this case, metric data are forecasted in block 2013 by combining the trend estimate given in Equation (2a) and the stochastic process model with the smallest accumulated residual error as follows:

{tilde over (z)} _(n+1) =T _(n+1) +{circumflex over (z)} _(n+l) ^((m))  (29)

FIG. 33A shows a plot of example trendy, non-periodic metric data and forecasted metric data over a forecast interval. Jagged curve 3302 represents a non-seasonal sequence of metric data with an increasing trend over historical window 3304. At time stamp t_(n), a forecast is requested for a forecast interval 3306. The parameters of the trend estimate and the stochastic process models are computed from the sequence of metric data in the historical window 3306 as described above. Jagged dashed-line curve 3308 represents forecasted metric data computed using Equation (29) at lead times in the forecast interval 3306.

(2) Metric data in the historical window may be pulse wave metric data. In this case, metric data are forecasted in block 2013 by combining the trend estimate given in Equation (2a) with the stochastic process model AR(0), ARMA(0,0), or ARIMA(0,0) and the latest pulse width and period given by back-sliding histogram as described above with reference to FIG. 32A as follows:

$\begin{matrix} {{{\overset{\sim}{z}}_{n + l} = {T_{n + l} + a_{n + 1} + {S\left( {A,{pw},p} \right)}}}{where}{{S\left( {A,{pw},p} \right)} = \left\{ {{{\begin{matrix} A & {{t_{s} + {m\; p}} \leq t_{n + l} \leq {t_{s} + {m\left( {p + {pw}} \right)}}} \\ 0 & {t_{n + l} < {t_{s} + {m\; p\mspace{14mu} {and}\mspace{14mu} t_{s}} + {m\left( {p + {pw}} \right)}} < t_{n + l}} \end{matrix}m} = 1},2,{{{\ldots \mspace{14mu} {for}\mspace{14mu} t_{s}} + {m\; p}} \leq {t_{n + L}.}}} \right.}} & (30) \end{matrix}$

FIG. 33B shows a plot of example trendy, pulse-wave metric data and forecasted metric data over a forecast interval. Pulses 3310-3313 represent sequence of pulse wave metric data over historical window 3314. At time stamp t_(n), a forecast is requested for a forecast interval 3316. Upward edges of forecasted pulses in the forecast interval 3314 are given by (t₅+mp, A+T_(n+1)+a_(n+1)) and downward edges of forecasted pulses in the forecast model are given by (t₅+m(p+pw), A+T_(n+l)+a_(n+l)). Dashed-line pulses 3318 and 3320 represent two forecasted pulses of metric data computed using Equation (30) over the forecast interval 3316.

(3) Metric data in the historical window may not have a pulse wave pattern but may have a seasonal period. In this case, metric data are forecasted in block 2013 by combining the trend estimate given in Equation (2a) with the stochastic process model AR(0), ARMA(0,0), or ARIMA(0,0) and the seasonal period model with the latest principle period P given by the back-sliding histogram as described above with reference to FIG. 32B as follows:

{tilde over (Z)} _(n+1) =T _(n+l) +a _(n+1) +S _((n+l)modP)  (31)

where

-   -   P is the principle period (i.e., P=N/k*); and     -   S_((n+l)mod P)={circumflex over (z)}_((n−P+1)mod P).

FIG. 33C shows a plot of example trendy, seasonal metric data and forecasted metric data over a forecast interval. Sinusoidal curve 3322 represents seasonal metric data over the historical window 3324. At time stamp t_(n), a forecast is requested for a forecast interval 3326. The parameters of the trend estimate and the seasonal periodic model are computed from the sequence of metric data in the historical window 3324 as described above. Dashed curve 3328 represents forecasted metric data computed using Equation (31) over the forecast interval 3326.

Returning to FIG. 17, the capacity planning application 1716 utilizes the analytics services manager 1702 to monitor each resource used by a virtual object by estimating the time remaining before each resource is projected to become overburdened based on forecasted metric data. Time remaining may be estimated for long term planning, such a 3 to 15 months. The capacity planning application 1716 also computes a projected right-size usable for each resource used by a virtual object based on the forecasted metric data.

The capacity planning application 1716 computes resource utilization metrics from the forecasted metric data {tilde over (z)}_(n+l) for each lead time t_(n+l) (i.e., l=1, 2, . . . , L) in the forecast interval. The resource utilization metric may be a mean forecast given by:

$\begin{matrix} {g_{n + l} = {\frac{1}{l}{\sum\limits_{j = 1}^{l}{\overset{\sim}{z}}_{n + j}}}} & (32) \end{matrix}$

an upper confidence value given by

u _(cn+l) ={tilde over (z)} _(n+l) +Cσ(l)  (33a)

or a lower confidence value given by

lc _(n+l) ={tilde over (z)} _(n+l) −Cσ(l)  (33b)

where

-   -   C is a prediction interval coefficient; and     -   σ(l) is an estimated standard deviation of the l-th lead time in         the forecast interval.         The resource utilization metric may be the linear trend T_(n+l)         with parameters computed as described above with reference to         Equations (2a)-(2d). The upper and lower confidence values may         be centered on the estimated liner trend T_(n+l):

uc _(n+l) =T _(n+l) +Cσ(l)  (34a)

lc _(n+l) =T _(n+l) −Cσ(l)  (34b)

The upper and lower confidence values are confidence bounds of a confidence interval denoted by [lc_(n+l), uc_(n+l)]. The prediction interval coefficient C corresponds to a probability that a metric data value produced by a resource in the forecast interval will lie in the confidence interval [lc_(n+l),uc_(n+l)]. Examples of prediction interval coefficients are provided in the following table:

Coefficient (C) Percentage (%) 2.58 99 1.96 95 1.64 90 1.44 85 1.28 80 0.67 50 For example, a 95% confidence gives a confidence interval [T_(n+l)−1.96σ(l), T_(n+l)+1.96σ(l)]. In other words, there is a 95% chance that the n+l-th metric data value of the resource will lie within the confidence interval based on the metric data in the historical window.

When forecasting is executed using one of the stochastic models, as described above with reference to Equation (29), the estimated standard deviation σ(l) in Equations (33a)-(34b) is given by:

$\begin{matrix} {{\sigma (l)} = \sqrt{\sigma_{a}^{2}{\sum\limits_{j = 1}^{l - 1}\psi_{j}^{2}}}} & (35) \end{matrix}$

where the ψ_(j)'s are weights.

The estimated standard deviation of Equation (35) increases parabolically as the lead time index l increases. The weights ψ_(j) in Equation (35) are recursively computed based on the autoregressive and moving-average weight parameters described above with reference to Equation (6a). When forecasting is executed using an AR model represented by Equation (12), the weights of Equation (35) are computed recursively as follows:

$\begin{matrix} {\psi_{j} = {\sum\limits_{i = 1}^{p}{\varphi_{i}\psi_{j - i}}}} & \left( {36a} \right) \end{matrix}$

where ψ₀=1.

When forecasting is executed using an ARMA model represented by Equation (6a), the weights of Equation (35) are computed recursively as follows:

$\begin{matrix} {\psi_{j} = {\sum\limits_{i = 1}^{p}{\varphi_{i}\psi_{j - i}\theta_{j}}}} & \left( {36b} \right) \end{matrix}$

-   -   where θ_(j)=0 for j>q.         When forecasting is executed using an ARIMA model represented by         Equation (13), the weights of Equation (35) are computed         recursively as follows:

$\begin{matrix} {\psi_{j} = {{\sum\limits_{i = 1}^{p + d}{\varphi_{i}\psi_{j - i}}} - \theta_{j}}} & \left( {36c} \right) \end{matrix}$

In another implementation, when forecasting is executed using one of the stochastic models or the seasonal period model, as described above with reference to Equations (29) and (31), the estimated standard deviation σ(l) may be a residual standard deviation computed as follows:

$\begin{matrix} {{\sigma (l)} = {\sigma = \sqrt{\frac{1}{n - 1}{\sum\limits_{i = 1}^{n}\left( {{\overset{\sim}{z}}_{n} - {\hat{z}}_{n}} \right)^{2}}}}} & (37) \end{matrix}$

where {tilde over (z)}_(n) are forecasted metric data values computed using Equation (29) or (31) in the historical window.

The residual standard deviation of Equation (37) is constant as the lead time index l increases.

In another implementation, the estimated standard deviation σ(l) is computed as follows:

σ(l)=σ√{square root over (l)}  (38)

The confidence interval of Equation (38) parabolically increases as the lead time index l increases.

FIGS. 34A-34D show example plots of confidence intervals computed for the same set of forecast metric data. Horizontal axes, such as horizontal axis 3402, represent time. Vertical axes, such as vertical axis 3404, represent metric data value amplitudes. Solid dots, such as solid dot 3406, represent metric data values in a historical window. Open dots, such as open dot 3408, represent forecasted metric data values. The plots include shaded regions that illustrate the magnitudes of the different confidence intervals. In the example of FIG. 34A, bars centered on the forecasted metric data, such as bar 3410 centered on forecasted metric data value 3408, represent confidence intervals computed using Equations (33a)-(33b) with the estimated standard deviation compute according to Equation (35). In the example of FIG. 34B, dashed line 3412 represents an increasing linear trend in the metric data computed as described above with reference to Equations (2a)-(2d). Bars centered on the linear trend 3412 at lead times, such as bar 3414, represent confidence intervals computed using Equations (34a)-(34b) with the estimated standard deviation compute according to Equation (35). Note that in FIGS. 34A-34B, the magnitudes of the confidence intervals parabolically increase with increasing lead time index l. In the example of FIG. 34C, bars centered on the linear trend 3412 at lead times, such as bar 3416, represent confidence intervals computed using Equations (34a)-(34b) with the constant estimated standard deviation compute according to Equation (37). In the example of FIG. 34D, shaded region 34 illustrates parabolically increasing confidence intervals that follow the seasonal pattern of the forecasted metric data values in the forecast interval.

Each resource of a distributed computing system has an associated usable capacity given by:

UC=TC−R  (39)

where

-   -   TC is the total physical capacity of the resource; and     -   R is the reserve capacity of the resource.         The usable capacity is the amount of the total physical capacity         a virtual object is allowed to use after setting aside the         reserve capacity. For example, suppose the total physical         capacity of memory is 128 GB and 18 GB of the memory is set         aside as reserve memory. The usable capacity available for         running a virtual object is 110 GB. As another example, a CPU         reservation holds a set number of clock cycles in reserve. The         usable capacity of the CPU by a virtual object is the number of         clock cycles the CPU is configured to process minus the reserved         clock cycles.

A user may select a resource utilization metric from the mean forecasts, linear trend, upper confidence values, or lower confidence values to determine an estimated time remaining. The estimated time remaining corresponds to a lead time in the forecast interval when the resource is expected to become overburdened. The particular choice of resource utilization metric enables a user to select a cautious, moderately cautious, or less cautious estimate of the time remaining. A cautious estimate has a lower risk of overestimating the time remaining before the usable capacity of the resource is reached and the resource becomes overburdened, such as selecting the upper confidence values. On the other hand, a less cautious estimate has a higher risk of overestimating the time remaining before the usable capacity of the resource is reached and the resource becomes overburdened, such as selecting the lower confidence values.

Let RU_(n+l) denote a resource utilization metric for determining an estimated time remaining before a resource used to run a virtual object becomes overburdened. A explained above, a user may select the resource utilization metric as one of the uc_(n+l), lc_(n+l), g_(n+l), and T_(n+l) described above. When the following condition is satisfied

RU _(n+l) ≥UC  (40a)

for some lE [1, 2, . . . , L],

the resource is expected to be overburdened at an estimated time corresponding to a lead time t_(n+l) in the forecast interval. The estimated time remaining is given by

TR=t _(n+l) −t _(n)  (40b)

The estimated time remaining TR may be reported by generating an alert and displaying the estimated time remaining before the resource becomes overburdened in a graphical user interface on a system administrator display.

In addition to generating and displaying an alert, methods include executing automated or manual remedial measures prior to or at the estimated time t_(n+l) when the resource is projected to be overburdened. Let s denote a time less than the duration of the estimated time remaining TR. Remedial measures may be executed to prevent overburdening a resource when a current time t_(m) in the forecast interval satisfies the following condition:

t _(n+l) −ε≤t _(m)  (40c)

where 0≤ε<TR.

The remedial measures that may be automatically or manually executed to ensure the usable capacity of a resource used by a virtual object is sufficient in the forecast interval include:

-   -   1) decreasing the amount of the reserved capacity of the         resource, which increases the usable capacity of the resource,     -   2) in addition to the current resource used by the virtual         object, assigning one or more additional resources of the same         type to the virtual object,     -   3) migrating the virtual object to a different server computer         with the same type of resource but with a larger usable capacity         than the resource currently used by the virtual object, and     -   4) creating one or more additional virtual objects from a         template of the virtual object, the additional virtual objects         to share the workload of the virtual object.

FIG. 35A shows an example plot of a linearly increasing stochastic sequence of metric data, and FIG. 35B shows an example plot of an increasing seasonally varying sequence of metric data. The resources represented in FIGS. 35A and 35B may be CPU, memory, a storage drive, network throughput, or another resource of a server computer. Horizontal axes 3502 and 3504 represent time. Vertical axes 3506 and 3508 represent a range of amplitudes of metric data values. Line segments located along the time axes represent time stamps separated by a selected unit of time, such as seconds, minutes, an hour, six hours, twelve hours, a day, a week or longer. Metric data values are represented by solid dots with time stamps in historical windows. Each metric data value z_(i) may be a maximum, minimum, or an average of metric data values generated by the resource in a time interval (t_(i−1), t_(i)] or be a current metric data value at a corresponding time stamp t_(i). For example, each metric data value z_(i) may be a maximum CPU usage in a 24-hour time interval (t_(i)−24 hrs, t_(i)] and each time stamp t_(i) denotes a day. Forecasted metric data values in forecast intervals are represented by open dots.

In the example of FIG. 35A, upper confidence values uc_(n+l) represented by solid curve 3510 are a resource utilization metric for forecasted metric data values in the forecast interval. A usable capacity for the resource is denoted by dot-dashed line 3512. The upper confidence values 3510 exceed the usable capacity 3512 at lead time t_(n+l) (i.e., uc_(n+l) ≥UC), as identified by dashed line 3514. A cautious estimated time remaining is computed according to Equation (40a) and is denoted by directional arrow 3516. Using the condition represented by Equation (40b), when a time stamp of a metric data value exceeds t_(n+l) −ε, any of the remedial measures described above with reference to Equation (40b) may be executed to increase the usable capacity of the resource used by the virtual object, such as reducing the reserved capacity of the resource, which increases the usable capacity represented by dashed line 3518. FIG. 35A also shows a dotted line 3520 that represents a linear trend T_(n+l) extended into the forecast interval. The linear trend exceeds the usable capacity 3512 at lead time identified by dashed line 3522. Equations (40a) and (40b) can be used to compute an estimated time remaining represented by directional arrow 3524. Dashed curve 3526 represents lower confidence values lc_(n+l) for forecasted metric data values that exceed the usable capacity at some point in the future. Note that, in this example, if the forecasted metric values prove to be an accurate estimate of the increasing trend in resource usage, selection of the upper confidence values 3510 avoids the risk of overburdening the resource. Selection of the linear trend 3520 has a modest risk of allowing the resource to become overburdened. Selection of the lower confidence values 3526 increases the risk of overburdening the resource.

In the example of FIG. 35B, upper confidence values uc_(n+l) are represented by solid curve 3528 and lower confidence values lc_(n+l) are represented by dashed curve 3530. A usable capability for the resource is denoted by dot-dashed line 3532. An upper confidence value that exceeds the usable capacity 3532 at time stamp t_(n+l) (i.e., uc_(n+l) ≥UC) is identified by dashed line 3534. Equation (40a) can be used to compute a conservative estimated time remaining before the resource is expected to be overburdened, as denoted by directional arrow 3536. Using the condition represented by Equation (40b), when a time stamp of a metric data value exceeds t_(n+l) −ε, any of the remedial measures described above with reference to Equation (40b) may be executed to increase the usable capacity of the resource used by the virtual object, such as reducing the reserved capacity of the resource to increase the usable capacity represented by dashed line 3538. When a lower confidence value exceeds the usable capacity 3532, as indicated by dashed line 3540, equation (40a) can be used to compute a less cautious estimated time remaining before the resource is expected to be overburdened, as denoted by directional arrow 3542. When a time stamp satisfies the condition in Equation (40b), the above described remedial measures may be executed.

Methods include determining an estimated capacity remaining for a resource in the forecast interval. The capacity remaining is given by:

$\begin{matrix} {{{CR} = {{UC} - {RU}_{n + \hat{l}}}}{where}} & \left( {41a} \right) \\ {{RU}_{n + \hat{l}} = {\max\limits_{1 \leq l \leq L}\left\{ {RU}_{n + l} \right\}}} & \left( {41b} \right) \end{matrix}$

for some {circumflex over (l)}ϵ[1, 2, . . . , L].

The resource utilization metric may be selected to obtain a cautious, moderately cautious, or less cautious estimated capacity remaining at a lead time in the forecast interval. When the capacity remaining is greater than zero (i.e., CR>0) for lead times of the forecast interval, the amount of the resource projected to be used by the virtual object is sufficient for the duration of the forecast interval. On the other hand, when the capacity remaining is less than or equal to zero (i.e., CR≤0) at a lead time t_(n+l) in the forecast interval, the amount of the resource projected to be used by the virtual object is insufficient and remedial measures are executed to avoid not having enough of the resource available for the virtual object. The remedial measures include determining a right-size usable capacity for the resource used by the virtual object as follows:

UC _(RS) =UC+RU _(n+{circumflex over (l)})  (42)

The right-size usable capacity UC_(RS) may be used as a minimum threshold for increasing the usable capacity of a resource. One or more of the following remedial measures may be taken to ensure that the usable capacity of a resource used by a virtual object is sufficient for the duration of the forecast interval include:

-   -   1) increasing the amount of the usable capacity of the resource         to at least UC_(RS),     -   2) in addition to the current resource used by the virtual         object, assigning one or more additional resources of the same         type to the virtual object that collectively provide at least         UC_(RS) as a right-size usable capacity,     -   3) migrating the virtual object to a different server computer         with the same type of resource but with the right-size usable         capacity at least UC_(RS), and     -   4) creating one or more additional virtual objects from a         template of the virtual object, the additional virtual objects         to share the workload of the virtual object and provide a usable         capacity of at least UC_(RS).

FIG. 36A shows an example plot of a linearly increasing stochastic sequence of metric data, and FIG. 36B shows an example plot of an increasing seasonally varying sequence of metric data. The resources may be CPU, memory, a storage drive, network throughput, or another resource of a server computer. Horizontal axes 3602 and 3604 represent time. Vertical axes 3606 and 3608 represent a range of amplitudes of metric data values. Line segments located along the time axes represent time stamps separated by a selected unit of time, such as, seconds, minutes, an hour, six hours, twelve hours, a day, or even longer. Metric data values with time stamps in historical windows are represented by solid dots and forecasted metric data values with time stamps in the forecast intervals are represented by open dots, as described above with reference to FIGS. 35A-35B.

In the example of FIG. 36A, upper confidence values represented by solid line 3610 are used as the resource utilization metric for forecasted metric data values in the forecast interval. A usable capacity for the resource is denoted by dotted-dashed line 3612. In this example, because the metric data values and the forecasted metric data values increase linearly along the linear trend, the estimated capacity remaining exceeds the usable capacity 3612 at the final lead time t_(n+L) of the forecast interval (i.e., uc_(n+L)≥UC). Equation (42) is used to determine a right-size usable capacity for the resource and any one of the remedial measures described above with reference to Equation (42) may be executed to adjust the usable capacity to a right size for the resource used by the virtual object. For example, dashed line 3614 represents a right-size usable capacity for the resource over the forecast interval obtained by adding the maximum upper confidence value to the current usable capacity.

In the example of FIG. 36B, upper confidence values represented by solid curve 3616 are used as the resource utilization metric for forecasted metric data values in the forecast interval. A usable capacity for the resource is denoted by dotted-dashed line 3618. In this example, the maximum resource utilization occurs at lead time stamp t_(n+{circumflex over (l)}) and exceeds the usable capacity 3612 of the forecast interval (i.e., uc_(n+{circumflex over (l)}) ≥UC). Equation (42) is used to determine a right-size usable capacity for the resource and any one of the remedial measures described above with reference to Equation (42) may be executed to adjust the usable capacity to the right-size usable capacity of the resource used by the virtual object. For example, dashed line 3620 represents a right-size usable capacity for the resource over the forecast interval by adding the maximum upper confidence value to the current usable capacity.

Rather than increasing the usable capacity in a single step as represented in FIGS. 36A-36B, the right-size usable capacity may be incrementally increased in two or more steps over sub-intervals of the forecast interval. The right-size usable capacity over each sub-interval of the forecast interval may be computed by adding the upper confidence value at the end of each sub-interval to the linear trend as follows:

UC _(RS) =T _(Round(ξL/η)) +uc _(n+Round(ξL/η))  (43a)

where

-   -   η is the number of sub-intervals of the forecast interval;     -   ξ=1, . . . , η; and     -   Round(·) is a nearest integer function.         In an alternative implementation, the right-size usable capacity         over each sub-interval of the forecast interval may be computed         by adding the lower confidence value at the end of each         sub-interval to the linear trend as follows:

UC _(RS) =T _(Round(ξL/η)) +lc _(n+Round(ξL/η))  (43b)

FIG. 37 shows an example of incrementally increasing the right-size usable capacity of the resource represented in FIG. 36A in three steps over the forecast interval. In this example, the forecast interval is divided into three sub-intervals (i.e., η=3). Shaded squares 3702-3704 represent upper confidence values computed at the end of corresponding sub-intervals 3706-3708. The right-size usable capacity for each sub-interval is given by Equation (43a), as represented by line segments 3710-3712. At time t_(n), the right-size usable capacity of the resource is increased to T_(Round(L/3))+uc_(n+Round(L/3)) over the sub-interval [t_(n), t_(n+Round(L/3η))). At time t_(n+Round(L/3η)), the right-size usable capacity of the resource is increased to T_(Round(2L/3))+uc_(n+Round(2L/3)) over the sub-interval [t_(n+Round(L/3η))) t_(n+Round(2L/3η))). At time t_(n+Round(2L/3η)), the right-size usable capacity of the resource is increased to T_(n+L) UC_(n+L) over the sub-interval [t_(n+Round(2L/3η)), t_(n+L)).

The methods described below with reference to FIGS. 38-48 are stored in one or more data-storage devices as machine-readable instructions that when executed by one or more processors of the computer system shown in FIG. 1 to estimate time remaining before distributed computing resources become overburdened and determine right-size of usable capacities of distributed computing resources.

FIG. 38 shows a control-flow diagram of a method to estimate time remaining and determine right-size for usable capacity of a resource of a distributed computing system. In block 3801, a stream of metric data is received at a metric processor as described above with reference to FIG. 17. In block 3802, a routine “remove trend from the stream” is called. In block 3803, a routine “compute stochastic process models” is called. In block 3804, a routine “compute periodic models” is called. In decision block 3805, when a forecast requested is received, control flows to block 3806. Otherwise, control flows to block 3801. In block 3806, a routine “compute forecast” is called. In block 3807, a routine “estimate time remaining and right-size for usable capacity of resources” is called. In decision block 3808, when a user selects stop forecast, the analytics services manager stops sending the stream of metric data to the metric processor.

FIG. 39 shows a control-flow diagram of the routine “remove trend from the stream” called in block 3802 of FIG. 38. In block 3901, least squares parameters for the sequence of metric data in the historical window, as described above with reference to Equations (2c) and (2d). In block 3902, a goodness-of-fit parameter is computed as described above with reference to Equation (3). In decision block 3903, when the goodness-of-parameter is greater than a threshold, control flows to block 3904. In block 3904, a trend computed using the least squares parameters is subtracted from the metric data in the historical window, as described above with reference to Equations (2a), (4) and FIGS. 19B and 19C.

FIG. 40 shows a control-flow diagram of the routine “compute stochastic process models” called in block 3803 of FIG. 38. A loop beginning with block 4001 repeats the computational operations represented by blocks 4002-4007 for each J different stochastic models, where J is the number of different stochastic models. In block 4002, weight parameters of a stochastic process model are computed based on previous values of the non-trendy metric data in the historical window, as described above with reference to FIG. 20. In block 4003, when a new non-trendy (e.g., detrended) metric data values is received, estimated metric data values are computed using each of the stochastic process models as described above with reference to Equation (9) and FIG. 21. In block 4004, a residual error is computed for each of the stochastic process models as described above with reference to Equation (10). In block 4005, an accumulated residual error is computed for the stochastic model as described above with reference to Equation (10). In decision block 4006, when weight parameters and accumulated residual errors have been computed for each of stochastic process models, control flow to block 4008. Otherwise, the parameter j is incremented in block 4907. In block 4008, a minimum residual error is initialized (e.g., Error(s)=100). A loop beginning with block 4009 repeats the computational operations of blocks 4010-4012 for each stochastic process model to identify the stochastic process model with the smallest accumulated residual error. In decision block 4010, when the accumulated residual error of the j-th stochastic process model is less the minimum residual error, control flow to block 4011. Otherwise, control flows to decision block 4012. In block 4011, the minimum residual error is set equal to the accumulated residual error. In decision block 4012, when accumulated residual errors for all J of the stochastic residual models have been considered control returns to FIG. 38. In block 4013, the parameter j is incremented.

FIG. 41 shows a control-flow diagram of the routine “compute periodic models” called in block 3804 of FIG. 38. In block 4101, logical parameters “Pulse wave” and “Seasonal” are set to FALSE. In block 4102, the EWMA is computed as described above with reference to Equation (14a). In decision block 4103, when the absolute difference Δ_(n+1) satisfies the condition given by Equation (14b), control flows to block 4104. Otherwise, control flows to block 4106. In block 4104, a routine “apply edge detection” is called. If pulse edges are determined in block 4104, “Pulse wave” is set to TRUE as described below with reference to FIG. 42. In decision block 4105, if “Pulse wave” is set to TRUE, control flows to block 4104. Otherwise, control flows to block 4106. In block 4106, a routine “compute seasonal parameters” is called. If a seasonal parameter is determined in block 4106, “Seasonal” is set to TRUE as described below with reference to FIG. 43.

FIG. 42 shows a control-flow diagram of the routine “apply edge detection” called in block 4104 of FIG. 41. In block 4201, an average of metric data values in a previous time window is computed as described above with reference to Equation (15a). In block 4202, an average of metric data values in a previous time window is computed as described above with reference to Equation (15b). In block 4203, a change parameter is computed based on an absolute difference between the averages of metric data values as described above with reference to Equation 16. In decision block 4204, when the change parameter is greater than a maximum amplitude range threshold computed as described above with reference to Equation (17a), control flows to block 4205. Otherwise, control flows to decision block 4211. In block 4205, an EWMA of change parameters is computed with the change parameter computed in block 4203 as the most recent change parameter. In decision block 4206, when the change parameter is greater than an EWMA threshold as described above with reference to Equation (17b), control flows to block 4207. Otherwise, control flows to decision block 4211. In block 4207, a largest absolute difference between consecutive pairs of metric data values is determined for metric data values in a time interval formed from the previous and next time windows as described above with reference to Equation (18). In block 4208, when the metric data values of the largest absolute difference between consecutive metric data values is increasing (i.e., z_(i+1)−z_(i)>0), the edge is an upward edge and control flows to block 4210. Otherwise, the largest absolute difference between consecutive metric data values is decreasing (i.e., z_(i+1)−z_(i)<0), control flows to block 4209. In block 4209, a downward edge is recorded for time t_(i). In block 4210, an upward edge is recorded for time t_(i+1). In decision block 4211, when the duration of the time window of the historical window is complete control flows to decision block 4212. Otherwise, control flows to block 4213. In block 4212, the previous and next time windows are advanced as described above with reference to FIG. 27A and the average of metric data values in the next time window assigned to the average of metric data values of the previous time window. In decision block 4213, when upward and/or downward edges have been identified as described above with reference to blocks 4208-4210, control flows to block 4214. Otherwise, control flows back to the routine “compute periodic models” in FIG. 41. In block 4214, pulse width and period of a pulse wave are computed as described above with reference to FIG. 28. In block 4215, pulse wave back-sliding histograms of pulse width and period are updated as described above with reference to FIG. 32A. In block 4216, “Pulse wave” is set to TRUE.

FIG. 43 shows a control-flow diagram of the routine “compute seasonal parameters” called in block 4104 of FIG. 41. A loop beginning with block 4301 repeats the computational operations represented by blocks 4302-4308 for smallest to largest sampling rate, as described above with reference to FIG. 29. In block 4302, metric data in the historical window is resampled according to the sample rate as described above with reference to FIG. 29. In block 4303, a routine “determine period of stream” is called. In decision block 4304, when a period for the metric data has been determined in block 4303, control flows to block 4305. In block 4305, “Seasonal” is set to TRUE. In decision block 4306, the computational operations represented by blocks 4302-4304 are repeated for another sample rate.

FIG. 44 shows a control-flow diagram of the routine “determine period of stream” called in block 4303 of FIG. 43. A loop beginning with block 4401 repeats the computational operations represented by blocks 4402-4413 for each short-time window as described above with reference to FIG. 30. In block 4402, a periodogram is computed for a short-time window of the historical window as described above with reference to Equations (19)-(20) and FIG. 30. In block 4403, the area of the periodogram is computed as described above with reference to Equation (21). In decision block 4404, if no frequencies of the periodogram satisfy the condition of Equation (22), then no candidate principle frequencies exist in the periodogram and the routine does not return a seasonal period for the short-time window. Otherwise, control flows to block 4413. In block 4405, a circular autocorrelation function is computed in neighborhoods of candidate periods that correspond to the candidate principle frequencies, as described above with reference to Equation (23) and FIG. 31B. A loop beginning with block 4406 repeats the computational operations of blocks 4407-4409 for each candidate period. In block 4407, curvature near a candidate period is estimated as described above with reference to Equation (25). In decision block 4408, when the curvature corresponds to a local maximum, control flows to block 4409. In block 4409, the period is refined to the period that corresponds to the maximum ACF value in the neighborhood and the period is returned as the principle period, as described above with reference to FIG. 31C. In decision block 4410, operations represented by blocks 4407-4409 are repeated for another candidate period. Otherwise, no principle period (i.e., principle frequency) is returned. In block 4411, a principle frequency back-sliding histogram stored in a circular buffer is updated as described above with reference to FIG. 32B. In block 4412, the frequency counts of the frequencies in the principle frequency back-sliding histogram are rank ordered from largest to smallest. In decision block 4413, when the short-time windows resampled historical window have been considered, control flows to decision block 4414. Otherwise, the computational operations represented by blocks 4402-4413 are repeated for another location of the short-time window in the historical window of resampled metric data. In decision block 4414, when the condition described above with reference to Equation (27) is satisfied, the principle period that corresponds to dominant principle frequency is returned.

FIG. 45 shows a control-flow diagram the routine “compute forecast” called in block 3806 of FIG. 38. In decision block 4501, when “Pulse wave” equals TRUE, control flows to block 4502. In block 4502, a forecast is computed over a forecast interval as described above with reference to Equation (30). In decision block 4503, when “Seasonal” equals TRUE, control flows to block 4504. In block 4504, a forecast is computed over the forecast interval as described above with reference to Equation (31). In block 4505, a forecast is computed over the forecast interval as described above with reference to Equation (29).

FIG. 46 shows a control-flow diagram of the routine “estimate time remaining and right-size for usable capacity of resource” called in block 3707 of FIG. 37. In decision block 4601, if a user decides to determine estimate time remaining for a resource, control flows to block 4602. Otherwise, a user may decide to determine a right-size usable capacity of the resource and control flows to block 4605. For example, a systems administrator may be presented with the two options in a graphical user interface in which the systems administrator may select one of the two options. In block 4602, a routine “determine estimated time remaining for resource” is called. The routine “determine estimated time remaining for resource” projects that the resource will become overburden in the forecast interval as described below with reference to FIG. 47, a logical variable “overburdened” is set to TRUE. In decision block 4603, when the logical variable “overburdened” is set to TRUE in block 4602, control flows to block 4604. In block 4604, an alert may be generated and displayed on the graphical user interface and the system administrator may select to execute any one or more of the remedial measures described above with reference to Equation (40c). Alternatively, one or more of the remedial measures described above with reference to Equation (40c) may be automatically executed. In block 4605, a routine “determine right size for usable capacity of resource” is called. The routine “determine right size for usable capacity of resource” projects that the resource will exceed the usable capacity of the resource in the forecast interval as described below with reference to FIG. 48, a logical variable “no capacity remaining” is set to TRUE. In decision block 4606, when the logical variable “no capacity remaining” is set to TRUE in block 4605, control flows to block 4607. In block 4607, an alert may be generated and displayed on the graphical user interface and the system administrator may select to execute any one or more of the remedial measures described above with reference to Equation (42). Alternatively, one or more of the remedial measures described above with reference to Equation (42) may be automatically executed.

FIG. 47 shows a control-flow diagram of the routine “determine estimated time remaining for resource” called in block 4602 of FIG. 46. In block 4701, the logical variable “overburdened” is set to FALSE. A loop beginning with block 4702 repeats the computational steps of blocks 4703-4706 for each lead time index. In block 4703, a resource utilization metric is computed for each lead time in the forecast interval as described above with reference to Equations (32)-(38). The resource utilization metric may be selected by the user. In decision block 4704, when the resource utilization metric is greater than the usable capacity at a lead time in the forecast interval, control flows to block 4705. In block 4705, the logical variable “overburdened” is set to TRUE and the routine returns controls to the routine in FIG. 46. In decision block 4706, if the lead time index l equals L, the routine returns control to FIG. 46.

FIG. 48 shows a control-flow diagram of the routine “determine right size for usable capacity of resource” called in block 4605 of FIG. 46. In block 4801, a logical variable “no capacity remaining” is set to FALSE. A loop beginning with block 4802 repeats the computational steps of block 4803-4807 for each lead time index. In block 4803, a resource utilization metric is computed for each lead time in the forecast interval as described above with reference to Equations (32)-(38). The resource utilization metric may be selected by the user. In block 4804, capacity remaining for the resource is computed as described above with reference to Equations (41a)-(41b). In decision block 4805, if the capacity remaining is less than zero, control flows to block 4806. In block 4806, the logical variable “no capacity remaining” is set to TRUE. In decision block 4804, if the lead time index I equals L, the routine returns control to FIG. 46.

It is appreciated that the previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present disclosure. Various modifications to these embodiments will be apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the disclosure. Thus, the present disclosure is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein. 

1. A process stored in one or more data-storage devices and executed using one or more processors of a computer system to adjust usable capacity of a resource used by virtual objects of a distributed computing system, the process comprising: computing two or more models for forecasting metric data based on a sequence of metric data that represent usage of the resource in a historical window; selecting a model of the two or more models that corresponds to the sequence of metric data in the historical window; computing forecasted metric data in a forecast interval using the model; computing a resource utilization metric based on the forecasted metric data; when the resource utilization metric is greater than the usable capacity of the resource, determining an estimated time remaining in the forecast internal that corresponds to an estimated time in the forecast interval when the usable capacity of the resource is expected to be insufficient; and adjusting the usable capacity of the resource used by the virtual objects when the resource is projected to have insufficient usable capacity in the forecast interval.
 2. The method of claim 1, wherein computing the two or more models for forecasting metric comprises: detrending a sequence of metric data to obtain a sequence of non-trendy metric data, the sequence of metric data representing latest usage of a resource in the distributed computing system; computing stochastic process models of the sequence of non-trendy metric data and corresponding accumulated residual errors; computing a pulse wave model based on changes in averages of metric data in two sliding adjacent time windows of the sequence of non-trendy metric data; and computing two or more seasonal models for the sequence of non-trendy metric data for two or more sample rates applied to the sequence of non-trendy metric data.
 3. The process of claim 1 wherein computing the sequence of forecasted metric data over the forecast interval comprises computing the sequence of forecasted metric data over the forecast interval at each lead time of the forecast interval based on the model.
 4. The process of claim 1 wherein computing the resource utilization metric comprises: computing an estimated standard deviation at each lead time of the forecast interval; and computing the resource utilization metric as an upper confidence value based on the estimated standard deviation and a prediction interval coefficient.
 5. The process of claim 1 wherein computing the resource utilization metric comprises: computing an estimated standard deviation at each lead time of the forecast interval; and computing the resource utilization metric as a lower confidence value based on the estimated standard deviation and a prediction interval coefficient.
 6. The process of claim 1 wherein computing the resource utilization metric comprises computing the resource utilization metric as a mean of forecasted metric data.
 7. The process of claim 1 wherein adjusting the usable capacity of the resource comprises executing one or more of decreasing a reserved capacity of the resource to increases the usable capacity of the resource; assigning one or more additional resources of the same type to the virtual object; migrating the virtual object to a different server computer with the same type of resource but with a larger usable capacity than the resource currently used by the virtual object; and creating one or more additional virtual objects from a template of the virtual object, the additional virtual objects to share the workload of the virtual object.
 8. A computer system to adjust usable capacity of a resource used by virtual objects of a distributed computing system, the system comprising: one or more processors; one or more data-storage devices; and machine-readable instructions stored in the one or more data-storage devices that when executed using the one or more processors controls the system to carry out computing two or more models for forecasting metric data based on a sequence of metric data that represents usage of the resource in a historical window; selecting a model of the two or more models that corresponds to the sequence of metric data in the historical window; computing forecasted metric data in a forecast interval using the model; determining a capacity remaining for the resource based on the forecasted metric data; and when the capacity remaining is less than or equal to zero, computing a right-size usable capacity for the resource and adjusting the usable capacity of the resource to at least the right-size usable capacity.
 9. The computer system of claim 8 wherein computing the two or more models for forecasting metric comprises: detrending a sequence of metric data to obtain a sequence of non-trendy metric data, the sequence of metric data representing latest usage of a resource in the distributed computing system; computing stochastic process models of the sequence of non-trendy metric data and corresponding accumulated residual errors; computing a pulse wave model based on changes in averages of metric data in two sliding adjacent time windows of the sequence of non-trendy metric data; and computing two or more seasonal models for the sequence of non-trendy metric data for two or more sample rates applied to the sequence of non-trendy metric data.
 10. The computer system of claim 8 wherein computing the sequence of forecasted metric data over the forecast interval comprises computing the sequence of forecasted metric data over the forecast interval at each lead time of the forecast interval based on the model.
 11. The computer system of claim 8 wherein computing the resource utilization metric comprises: computing an estimated standard deviation at each lead time of the forecast interval; and computing the resource utilization metric as an upper confidence value based on the estimated standard deviation and a prediction interval coefficient.
 12. The computer system of claim 8 wherein computing the resource utilization metric comprises: computing an estimated standard deviation at each lead time of the forecast interval; and computing the resource utilization metric as a lower confidence value based on the estimated standard deviation and a prediction interval coefficient.
 13. The computer system of claim 8 wherein computing the resource utilization metric comprises computing the resource utilization metric as a mean of forecasted metric data.
 14. The computer system of claim 8 wherein adjusting the usable capacity of the resource comprises executing one or more of increasing the amount of the usable capacity of the resource to at least the right-size usable capacity, assigning one or more additional resources of the same type to the virtual object that collectively provide at least the right-size usable capacity, migrating the virtual object to a different server computer with the same type of resource but with at least the right-size usable capacity, and creating one or more additional virtual objects from a template of the virtual object, the additional virtual objects to share the workload of the virtual object and provide at least the right-size usable capacity.
 15. A non-transitory computer-readable medium encoded with machine-readable instructions that implement a method carried out by one or more processors of a computer system to perform the operations of computing two or more models for forecasting metric data based on a sequence of metric data generated by a resource used to virtual objects in a distributed computing system; selecting a model of the two or more models that corresponds to the sequence of metric data in the historical window; computing forecasted metric data in a forecast interval using the model; determining a capacity remaining for the resource based on the forecasted metric data; and when the capacity remaining is less than or equal to zero, computing a right-size usable capacity for the resource and adjusting the usable capacity of the resource to at least the right-size usable capacity.
 16. The medium of claim 15 wherein computing the two or more models for forecasting metric comprises: detrending a sequence of metric data to obtain a sequence of non-trendy metric data, the sequence of metric data representing latest usage of a resource in the distributed computing system; computing stochastic process models of the sequence of non-trendy metric data and corresponding accumulated residual errors; computing a pulse wave model based on changes in averages of metric data in two sliding adjacent time windows of the sequence of non-trendy metric data; and computing two or more seasonal models for the sequence of non-trendy metric data for two or more sample rates applied to the sequence of non-trendy metric data.
 17. The medium of claim 15 wherein computing the sequence of forecasted metric data over the forecast interval comprises computing the sequence of forecasted metric data over the forecast interval at each lead time of the forecast interval based on the model.
 18. The medium of claim 15 wherein computing the resource utilization metric comprises: computing an estimated standard deviation at each lead time of the forecast interval; and computing the resource utilization metric as an upper confidence value based on the estimated standard deviation and a prediction interval coefficient.
 19. The medium of claim 15 wherein computing the resource utilization metric comprises: computing an estimated standard deviation at each lead time of the forecast interval; and computing the resource utilization metric as a lower confidence value based on the estimated standard deviation and a prediction interval coefficient.
 20. The medium of claim 15 wherein computing the resource utilization metric comprises computing the resource utilization metric as a mean of forecasted metric data.
 21. The medium of claim 15 wherein adjusting the usable capacity of the resource comprises executing one or more of increasing the amount of the usable capacity of the resource to at least the right-size usable capacity, assigning one or more additional resources of the same type to the virtual object that collectively provide at least the right-size usable capacity, migrating the virtual object to a different server computer with the same type of resource but with at least the right-size usable capacity, and creating one or more additional virtual objects from a template of the virtual object, the additional virtual objects to share the workload of the virtual object and provide at least the right-size usable capacity. 